| L(s) = 1 | + 4-s + 2·7-s + 3·9-s + 12·11-s + 18·13-s + 6·17-s − 12·19-s + 25-s + 2·28-s + 3·36-s − 2·37-s + 2·41-s + 12·44-s − 20·47-s + 12·49-s + 18·52-s − 6·53-s − 30·59-s − 12·61-s + 6·63-s − 64-s − 16·67-s + 6·68-s − 4·71-s + 12·73-s − 12·76-s + 24·77-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 0.755·7-s + 9-s + 3.61·11-s + 4.99·13-s + 1.45·17-s − 2.75·19-s + 1/5·25-s + 0.377·28-s + 1/2·36-s − 0.328·37-s + 0.312·41-s + 1.80·44-s − 2.91·47-s + 12/7·49-s + 2.49·52-s − 0.824·53-s − 3.90·59-s − 1.53·61-s + 0.755·63-s − 1/8·64-s − 1.95·67-s + 0.727·68-s − 0.474·71-s + 1.40·73-s − 1.37·76-s + 2.73·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.496216856\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.496216856\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 - p T^{2} )^{2}( 1 + p T^{2} + p^{2} T^{4} ) \) |
| 7 | $D_4\times C_2$ | \( 1 - 2 T - 8 T^{2} + 4 T^{3} + 67 T^{4} + 4 p T^{5} - 8 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$$\times$$C_2^2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} ) \) |
| 17 | $D_4\times C_2$ | \( 1 - 6 T + 48 T^{2} - 216 T^{3} + 1211 T^{4} - 216 p T^{5} + 48 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 12 T + 82 T^{2} + 408 T^{3} + 1707 T^{4} + 408 p T^{5} + 82 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 2106 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 12 T^{2} + 950 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 26 T^{2} + 1899 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 2 T - 67 T^{2} + 22 T^{3} + 3196 T^{4} + 22 p T^{5} - 67 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 98 T^{2} + 4899 T^{4} - 98 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 10 T + 116 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 + 6 T + 29 T^{2} - 594 T^{3} - 4668 T^{4} - 594 p T^{5} + 29 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 30 T + 484 T^{2} + 5520 T^{3} + 48075 T^{4} + 5520 p T^{5} + 484 p^{2} T^{6} + 30 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 12 T + 166 T^{2} + 1416 T^{3} + 13131 T^{4} + 1416 p T^{5} + 166 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 + 4 T - 22 T^{2} - 416 T^{3} - 4733 T^{4} - 416 p T^{5} - 22 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 6 T + 80 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^3$ | \( 1 + 14 T^{2} - 6045 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 24 T + 278 T^{2} + 3168 T^{3} + 34107 T^{4} + 3168 p T^{5} + 278 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 24 T + 414 T^{2} + 5328 T^{3} + 58451 T^{4} + 5328 p T^{5} + 414 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 16806 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.388129070485076458517396808410, −8.208913062469036266117288489359, −7.68560742990758757141084451408, −7.47451310520575622992258442464, −7.46807023511644957098392711735, −6.68043874320124796170149601140, −6.54775307480905646865314999165, −6.40357825123527458600253275971, −6.40015526654556680904284248397, −6.19442822768259124282430017006, −5.93794620217405966245438215087, −5.60661376064671103932181287861, −5.18319793104340302800978732786, −4.54904320935798573400354869141, −4.38959869367523672375465893895, −4.20813692592435040024906994706, −3.95363826294617695564859066828, −3.73085993349248452966674878099, −3.35443218889112282183248860207, −3.28551888779052966624214230986, −2.61898925174842273262566120877, −1.52340179355847969659158163645, −1.45396338373658972504584153833, −1.44104629430829639453692161371, −1.41888258846302516276256559244,
1.41888258846302516276256559244, 1.44104629430829639453692161371, 1.45396338373658972504584153833, 1.52340179355847969659158163645, 2.61898925174842273262566120877, 3.28551888779052966624214230986, 3.35443218889112282183248860207, 3.73085993349248452966674878099, 3.95363826294617695564859066828, 4.20813692592435040024906994706, 4.38959869367523672375465893895, 4.54904320935798573400354869141, 5.18319793104340302800978732786, 5.60661376064671103932181287861, 5.93794620217405966245438215087, 6.19442822768259124282430017006, 6.40015526654556680904284248397, 6.40357825123527458600253275971, 6.54775307480905646865314999165, 6.68043874320124796170149601140, 7.46807023511644957098392711735, 7.47451310520575622992258442464, 7.68560742990758757141084451408, 8.208913062469036266117288489359, 8.388129070485076458517396808410
