| L(s) = 1 | + 16·2-s + 247.·3-s + 256·4-s + 3.95e3·6-s − 2.40e3·7-s + 4.09e3·8-s + 4.13e4·9-s − 2.79e4·11-s + 6.32e4·12-s − 6.09e4·13-s − 3.84e4·14-s + 6.55e4·16-s + 3.58e5·17-s + 6.61e5·18-s − 3.91e5·19-s − 5.93e5·21-s − 4.47e5·22-s + 3.02e5·23-s + 1.01e6·24-s − 9.75e5·26-s + 5.35e6·27-s − 6.14e5·28-s + 6.73e6·29-s + 2.98e6·31-s + 1.04e6·32-s − 6.90e6·33-s + 5.74e6·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.76·3-s + 0.5·4-s + 1.24·6-s − 0.377·7-s + 0.353·8-s + 2.10·9-s − 0.575·11-s + 0.880·12-s − 0.591·13-s − 0.267·14-s + 0.250·16-s + 1.04·17-s + 1.48·18-s − 0.689·19-s − 0.665·21-s − 0.406·22-s + 0.225·23-s + 0.622·24-s − 0.418·26-s + 1.93·27-s − 0.188·28-s + 1.76·29-s + 0.580·31-s + 0.176·32-s − 1.01·33-s + 0.736·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(7.746609699\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.746609699\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 16T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 2.40e3T \) |
| good | 3 | \( 1 - 247.T + 1.96e4T^{2} \) |
| 11 | \( 1 + 2.79e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 6.09e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 3.58e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 3.91e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 3.02e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 6.73e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 2.98e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 3.49e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 3.43e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.45e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.76e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 2.39e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.20e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 7.23e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 8.70e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.19e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.67e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.85e7T + 1.19e17T^{2} \) |
| 83 | \( 1 - 3.83e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 7.21e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 6.73e8T + 7.60e17T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.891036051090739838553474405787, −8.958816494811093919316293634469, −7.972107374444127906329010421625, −7.36851827064527936830845027545, −6.19089457216132332563497521224, −4.79476656166497799385138610315, −3.88119375288719906035970377190, −2.83267571465240037716271992142, −2.41976651157186546984911692236, −1.00773499789087693251589739433,
1.00773499789087693251589739433, 2.41976651157186546984911692236, 2.83267571465240037716271992142, 3.88119375288719906035970377190, 4.79476656166497799385138610315, 6.19089457216132332563497521224, 7.36851827064527936830845027545, 7.972107374444127906329010421625, 8.958816494811093919316293634469, 9.891036051090739838553474405787
