| L(s) = 1 | + 2·2-s − 3-s + 3·4-s − 2·6-s + 2·7-s + 4·8-s − 4·9-s + 5·11-s − 3·12-s − 8·13-s + 4·14-s + 5·16-s + 2·17-s − 8·18-s − 5·19-s − 2·21-s + 10·22-s − 23-s − 4·24-s − 16·26-s + 6·27-s + 6·28-s − 15·29-s − 3·31-s + 6·32-s − 5·33-s + 4·34-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 3/2·4-s − 0.816·6-s + 0.755·7-s + 1.41·8-s − 4/3·9-s + 1.50·11-s − 0.866·12-s − 2.21·13-s + 1.06·14-s + 5/4·16-s + 0.485·17-s − 1.88·18-s − 1.14·19-s − 0.436·21-s + 2.13·22-s − 0.208·23-s − 0.816·24-s − 3.13·26-s + 1.15·27-s + 1.13·28-s − 2.78·29-s − 0.538·31-s + 1.06·32-s − 0.870·33-s + 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35402500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35402500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 17 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 5 T + 17 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 5 T + 43 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + T + 15 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 15 T + 113 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 3 T + 63 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 12 T + 90 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 9 T + 71 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 13 T + 127 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 23 T + 225 T^{2} + 23 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - T + 105 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 7 T + 29 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 12 T + 138 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 9 T + 93 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 20 T + 222 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 10 T + 58 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 14 T + 102 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 22 T + 310 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
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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
−7.68025596717138404569502714597, −7.47693672658199577941346405025, −7.10056745934665870772245469520, −6.80011028303553122176383203963, −6.23119034330708047476001835686, −6.14432335211873751152391847804, −5.52892980132262655508899735718, −5.50902965764064822626388379720, −4.95889168868934007363484674254, −4.84763838987341718571065117546, −4.11589129834275091779802929837, −4.11437928225405586276453462636, −3.52110922142221533197071352970, −3.11679292363264095560894688719, −2.56253092833570864268856129388, −2.26450993951198084445683474509, −1.65187705114761134466444637340, −1.42973049422007441265404802382, 0, 0,
1.42973049422007441265404802382, 1.65187705114761134466444637340, 2.26450993951198084445683474509, 2.56253092833570864268856129388, 3.11679292363264095560894688719, 3.52110922142221533197071352970, 4.11437928225405586276453462636, 4.11589129834275091779802929837, 4.84763838987341718571065117546, 4.95889168868934007363484674254, 5.50902965764064822626388379720, 5.52892980132262655508899735718, 6.14432335211873751152391847804, 6.23119034330708047476001835686, 6.80011028303553122176383203963, 7.10056745934665870772245469520, 7.47693672658199577941346405025, 7.68025596717138404569502714597
