| L(s) = 1 | + 14·5-s + 4·11-s + 108·13-s − 14·17-s − 92·19-s − 152·23-s + 125·25-s + 212·29-s + 144·31-s − 158·37-s + 780·41-s − 1.01e3·43-s − 528·47-s + 606·53-s + 56·55-s − 364·59-s − 678·61-s + 1.51e3·65-s − 844·67-s + 16·71-s + 422·73-s − 384·79-s + 1.09e3·83-s − 196·85-s + 1.19e3·89-s − 1.28e3·95-s − 3.00e3·97-s + ⋯ |
| L(s) = 1 | + 1.25·5-s + 0.109·11-s + 2.30·13-s − 0.199·17-s − 1.11·19-s − 1.37·23-s + 25-s + 1.35·29-s + 0.834·31-s − 0.702·37-s + 2.97·41-s − 3.60·43-s − 1.63·47-s + 1.57·53-s + 0.137·55-s − 0.803·59-s − 1.42·61-s + 2.88·65-s − 1.53·67-s + 0.0267·71-s + 0.676·73-s − 0.546·79-s + 1.44·83-s − 0.250·85-s + 1.42·89-s − 1.39·95-s − 3.14·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.308368610\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.308368610\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 14 T + 71 T^{2} - 14 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 4 T - 1315 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 54 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 14 T - 4717 T^{2} + 14 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 92 T + 1605 T^{2} + 92 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 152 T + 10937 T^{2} + 152 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 106 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 144 T - 9055 T^{2} - 144 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 158 T - 25689 T^{2} + 158 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 390 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 508 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 528 T + 174961 T^{2} + 528 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 606 T + 218359 T^{2} - 606 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 364 T - 72883 T^{2} + 364 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 678 T + 232703 T^{2} + 678 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 844 T + 411573 T^{2} + 844 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 422 T - 210933 T^{2} - 422 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 384 T - 345583 T^{2} + 384 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 548 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 1194 T + 720667 T^{2} - 1194 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 1502 T + p^{3} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.000180299755715829088376851785, −8.716513940347302091798096182462, −8.390050338117211424068514403469, −8.187958782572275638078661158058, −7.66176092919759085609032099807, −6.95532128416389802725733227029, −6.46381411334078012928968298372, −6.39334461444201393752614273754, −5.89016383269228181608208425200, −5.85178418090096910570265142813, −4.98732680325032748249532463194, −4.66046360095585425227876365261, −4.17001431250774838681421756755, −3.69206658365482886912808801825, −3.14596983837836529175548715100, −2.71661798743668879415446923587, −1.91467595197289096487872979518, −1.71262069853415783459814326454, −1.14173216445116802085574670016, −0.44946874433453663507770217237,
0.44946874433453663507770217237, 1.14173216445116802085574670016, 1.71262069853415783459814326454, 1.91467595197289096487872979518, 2.71661798743668879415446923587, 3.14596983837836529175548715100, 3.69206658365482886912808801825, 4.17001431250774838681421756755, 4.66046360095585425227876365261, 4.98732680325032748249532463194, 5.85178418090096910570265142813, 5.89016383269228181608208425200, 6.39334461444201393752614273754, 6.46381411334078012928968298372, 6.95532128416389802725733227029, 7.66176092919759085609032099807, 8.187958782572275638078661158058, 8.390050338117211424068514403469, 8.716513940347302091798096182462, 9.000180299755715829088376851785
