| L(s) = 1 | − 2-s + 4-s + 4.41·5-s − 8-s − 4.41·10-s + 11-s + 16-s + 2.37·17-s + 3.68·19-s + 4.41·20-s − 22-s − 4.73·23-s + 14.5·25-s + 6.52·29-s + 3.37·31-s − 32-s − 2.37·34-s + 7.68·37-s − 3.68·38-s − 4.41·40-s + 12.1·41-s + 3.06·43-s + 44-s + 4.73·46-s − 4.10·47-s − 14.5·50-s − 8·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.97·5-s − 0.353·8-s − 1.39·10-s + 0.301·11-s + 0.250·16-s + 0.576·17-s + 0.846·19-s + 0.988·20-s − 0.213·22-s − 0.986·23-s + 2.90·25-s + 1.21·29-s + 0.606·31-s − 0.176·32-s − 0.407·34-s + 1.26·37-s − 0.598·38-s − 0.698·40-s + 1.89·41-s + 0.467·43-s + 0.150·44-s + 0.697·46-s − 0.599·47-s − 2.05·50-s − 1.09·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.733495127\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.733495127\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 - 4.41T + 5T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 2.37T + 17T^{2} \) |
| 19 | \( 1 - 3.68T + 19T^{2} \) |
| 23 | \( 1 + 4.73T + 23T^{2} \) |
| 29 | \( 1 - 6.52T + 29T^{2} \) |
| 31 | \( 1 - 3.37T + 31T^{2} \) |
| 37 | \( 1 - 7.68T + 37T^{2} \) |
| 41 | \( 1 - 12.1T + 41T^{2} \) |
| 43 | \( 1 - 3.06T + 43T^{2} \) |
| 47 | \( 1 + 4.10T + 47T^{2} \) |
| 53 | \( 1 + 8T + 53T^{2} \) |
| 59 | \( 1 + 12.5T + 59T^{2} \) |
| 61 | \( 1 + 3.79T + 61T^{2} \) |
| 67 | \( 1 - 10.1T + 67T^{2} \) |
| 71 | \( 1 + 0.934T + 71T^{2} \) |
| 73 | \( 1 - 7.46T + 73T^{2} \) |
| 79 | \( 1 + 0.418T + 79T^{2} \) |
| 83 | \( 1 + 2.14T + 83T^{2} \) |
| 89 | \( 1 + 12.8T + 89T^{2} \) |
| 97 | \( 1 + 7T + 97T^{2} \) |
| 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
−7.82048329661434306264640076662, −6.89047811334123369187217331728, −6.23753135569880566731239478718, −5.89350464101882157989080128142, −5.15054066021276469903707438674, −4.28274634708536620049242908881, −2.97264549523442758591650112644, −2.53048902054383020550996333561, −1.55060619195043140951986694777, −0.966169303660121754363543190941,
0.966169303660121754363543190941, 1.55060619195043140951986694777, 2.53048902054383020550996333561, 2.97264549523442758591650112644, 4.28274634708536620049242908881, 5.15054066021276469903707438674, 5.89350464101882157989080128142, 6.23753135569880566731239478718, 6.89047811334123369187217331728, 7.82048329661434306264640076662
