| L(s) = 1 | − 8·2-s − 2.80·3-s + 64·4-s − 438.·5-s + 22.4·6-s − 512·8-s − 2.17e3·9-s + 3.50e3·10-s − 5.48e3·11-s − 179.·12-s + 4.00e3·13-s + 1.22e3·15-s + 4.09e3·16-s − 2.80e4·17-s + 1.74e4·18-s + 2.38e4·19-s − 2.80e4·20-s + 4.38e4·22-s − 7.37e4·23-s + 1.43e3·24-s + 1.13e5·25-s − 3.20e4·26-s + 1.22e4·27-s − 9.87e4·29-s − 9.83e3·30-s − 4.74e4·31-s − 3.27e4·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.0599·3-s + 0.5·4-s − 1.56·5-s + 0.0424·6-s − 0.353·8-s − 0.996·9-s + 1.10·10-s − 1.24·11-s − 0.0299·12-s + 0.505·13-s + 0.0940·15-s + 0.250·16-s − 1.38·17-s + 0.704·18-s + 0.797·19-s − 0.784·20-s + 0.877·22-s − 1.26·23-s + 0.0212·24-s + 1.45·25-s − 0.357·26-s + 0.119·27-s − 0.751·29-s − 0.0665·30-s − 0.286·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.3595475828\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3595475828\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2.80T + 2.18e3T^{2} \) |
| 5 | \( 1 + 438.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 5.48e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 4.00e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.80e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.38e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 7.37e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 9.87e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 4.74e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.00e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 4.89e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 2.99e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 9.62e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.83e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.45e4T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.02e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.96e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.34e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.50e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.77e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 1.57e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 8.79e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.03e7T + 8.07e13T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.15043744021923676757882961180, −11.32475460114429429542670059832, −10.61972106196050701948158244371, −8.960701929093093467703982139109, −8.105187144657851268848749814374, −7.29376980566323184818965534265, −5.67584696866550976714345429075, −3.99441708889574000202854532260, −2.58547959850692534262413210468, −0.39201926283105232149773630032,
0.39201926283105232149773630032, 2.58547959850692534262413210468, 3.99441708889574000202854532260, 5.67584696866550976714345429075, 7.29376980566323184818965534265, 8.105187144657851268848749814374, 8.960701929093093467703982139109, 10.61972106196050701948158244371, 11.32475460114429429542670059832, 12.15043744021923676757882961180
