| L(s) = 1 | − 6·3-s + 6·5-s + 27·9-s − 26·11-s − 96·13-s − 36·15-s − 78·17-s + 40·19-s − 22·23-s + 114·25-s − 108·27-s + 204·29-s + 96·31-s + 156·33-s + 504·37-s + 576·39-s + 102·41-s − 296·43-s + 162·45-s − 780·47-s + 468·51-s + 192·53-s − 156·55-s − 240·57-s + 212·59-s + 100·61-s − 576·65-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.536·5-s + 9-s − 0.712·11-s − 2.04·13-s − 0.619·15-s − 1.11·17-s + 0.482·19-s − 0.199·23-s + 0.911·25-s − 0.769·27-s + 1.30·29-s + 0.556·31-s + 0.822·33-s + 2.23·37-s + 2.36·39-s + 0.388·41-s − 1.04·43-s + 0.536·45-s − 2.42·47-s + 1.28·51-s + 0.497·53-s − 0.382·55-s − 0.557·57-s + 0.467·59-s + 0.209·61-s − 1.09·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 - 6 T - 78 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 26 T + 2494 T^{2} + 26 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 96 T + 5350 T^{2} + 96 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 78 T + 8314 T^{2} + 78 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 20 T + p^{3} T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 22 T + 16030 T^{2} + 22 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 102 T + p^{3} T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 96 T - 24386 T^{2} - 96 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 504 T + 163462 T^{2} - 504 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 102 T + 99666 T^{2} - 102 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 296 T + 94646 T^{2} + 296 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 780 T + 326046 T^{2} + 780 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 192 T + 197782 T^{2} - 192 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 212 T + 355942 T^{2} - 212 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 100 T + 370190 T^{2} - 100 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 212 T + 611414 T^{2} + 212 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 534 T + 746334 T^{2} - 534 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 1128 T + 1062430 T^{2} + 1128 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 468 T - 92834 T^{2} + 468 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 824 T + 536870 T^{2} + 824 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2118 T + 2285746 T^{2} - 2118 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 400 T + 214046 T^{2} + 400 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.309134339515903686182956295745, −8.020925128404049452836734314382, −7.65304245795070738597626079704, −7.15272227493015153272749073792, −6.76791638762154943905542383318, −6.57920403214972481715013647868, −6.14068497936181272114404459024, −5.54889687436081444478935091467, −5.39589298163870708551857900944, −4.66370359794100889338614953086, −4.61233679853267335025760416061, −4.49662025390580685457895394807, −3.44461150994819372520582020864, −2.94571380293260757707466393308, −2.35378771984906195243764292402, −2.29060285207794744930531709270, −1.33531859941937378928365961322, −0.903377867021384661796800276563, 0, 0,
0.903377867021384661796800276563, 1.33531859941937378928365961322, 2.29060285207794744930531709270, 2.35378771984906195243764292402, 2.94571380293260757707466393308, 3.44461150994819372520582020864, 4.49662025390580685457895394807, 4.61233679853267335025760416061, 4.66370359794100889338614953086, 5.39589298163870708551857900944, 5.54889687436081444478935091467, 6.14068497936181272114404459024, 6.57920403214972481715013647868, 6.76791638762154943905542383318, 7.15272227493015153272749073792, 7.65304245795070738597626079704, 8.020925128404049452836734314382, 8.309134339515903686182956295745
