| L(s) = 1 | + 2-s + 2·3-s − 2·4-s + 2·6-s + 7·7-s − 3·8-s + 3·9-s − 4·12-s + 8·13-s + 7·14-s + 16-s + 3·17-s + 3·18-s − 5·19-s + 14·21-s + 7·23-s − 6·24-s + 8·26-s + 4·27-s − 14·28-s + 15·29-s − 31-s + 2·32-s + 3·34-s − 6·36-s + 7·37-s − 5·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s − 4-s + 0.816·6-s + 2.64·7-s − 1.06·8-s + 9-s − 1.15·12-s + 2.21·13-s + 1.87·14-s + 1/4·16-s + 0.727·17-s + 0.707·18-s − 1.14·19-s + 3.05·21-s + 1.45·23-s − 1.22·24-s + 1.56·26-s + 0.769·27-s − 2.64·28-s + 2.78·29-s − 0.179·31-s + 0.353·32-s + 0.514·34-s − 36-s + 1.15·37-s − 0.811·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19580625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19580625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.99357071\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.99357071\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 59 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - p T + 25 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 8 T + 37 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 3 T + 25 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 5 T + 13 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 7 T + 57 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 15 T + 113 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 + T - 39 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 7 T + 75 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 5 T + 57 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 15 T + 139 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 117 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 13 T + 153 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 105 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 2 T + 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 5 T + 141 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - T + 165 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 3 T + 29 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
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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
−8.544005146226432272656922549514, −8.252419121729963466604057602243, −7.895568097799451998359497492915, −7.76775957618844491658984454780, −7.22783324838223369304979300390, −6.70631292813660061906677071970, −6.13713755902147174466985432352, −6.01550206942327817836940518226, −5.35983522794232273778698523343, −5.06420405368235833901953968529, −4.65393605379834373742175500608, −4.39233867330923636445790238695, −4.01903409892236909916379294999, −3.94890213495850414802107872142, −2.97597375969831960587284578216, −2.96426507038382580072113218299, −2.28438336394772667286516783471, −1.58433951670370372795516323328, −1.10115142098212647082118293732, −1.01092318833724316360588099792,
1.01092318833724316360588099792, 1.10115142098212647082118293732, 1.58433951670370372795516323328, 2.28438336394772667286516783471, 2.96426507038382580072113218299, 2.97597375969831960587284578216, 3.94890213495850414802107872142, 4.01903409892236909916379294999, 4.39233867330923636445790238695, 4.65393605379834373742175500608, 5.06420405368235833901953968529, 5.35983522794232273778698523343, 6.01550206942327817836940518226, 6.13713755902147174466985432352, 6.70631292813660061906677071970, 7.22783324838223369304979300390, 7.76775957618844491658984454780, 7.895568097799451998359497492915, 8.252419121729963466604057602243, 8.544005146226432272656922549514
