| L(s) = 1 | + 2.52·2-s + 4.35·4-s − 5-s + 3.01·7-s + 5.92·8-s − 2.52·10-s − 5.74·11-s + 13-s + 7.59·14-s + 6.22·16-s + 5.46·17-s + 5.89·19-s − 4.35·20-s − 14.4·22-s + 8.39·23-s + 25-s + 2.52·26-s + 13.1·28-s + 3.37·29-s − 4.29·31-s + 3.84·32-s + 13.7·34-s − 3.01·35-s − 4.29·37-s + 14.8·38-s − 5.92·40-s − 6.56·41-s + ⋯ |
| L(s) = 1 | + 1.78·2-s + 2.17·4-s − 0.447·5-s + 1.13·7-s + 2.09·8-s − 0.796·10-s − 1.73·11-s + 0.277·13-s + 2.02·14-s + 1.55·16-s + 1.32·17-s + 1.35·19-s − 0.972·20-s − 3.08·22-s + 1.74·23-s + 0.200·25-s + 0.494·26-s + 2.47·28-s + 0.626·29-s − 0.771·31-s + 0.679·32-s + 2.36·34-s − 0.509·35-s − 0.706·37-s + 2.41·38-s − 0.936·40-s − 1.02·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.653044248\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.653044248\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 2.52T + 2T^{2} \) |
| 7 | \( 1 - 3.01T + 7T^{2} \) |
| 11 | \( 1 + 5.74T + 11T^{2} \) |
| 17 | \( 1 - 5.46T + 17T^{2} \) |
| 19 | \( 1 - 5.89T + 19T^{2} \) |
| 23 | \( 1 - 8.39T + 23T^{2} \) |
| 29 | \( 1 - 3.37T + 29T^{2} \) |
| 31 | \( 1 + 4.29T + 31T^{2} \) |
| 37 | \( 1 + 4.29T + 37T^{2} \) |
| 41 | \( 1 + 6.56T + 41T^{2} \) |
| 43 | \( 1 + 0.654T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 - 6.03T + 53T^{2} \) |
| 59 | \( 1 - 7.55T + 59T^{2} \) |
| 61 | \( 1 - 6.02T + 61T^{2} \) |
| 67 | \( 1 - 5.69T + 67T^{2} \) |
| 71 | \( 1 - 2.00T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 + 6.16T + 83T^{2} \) |
| 89 | \( 1 - 2.25T + 89T^{2} \) |
| 97 | \( 1 - 6.24T + 97T^{2} \) |
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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
−7.898880420617599060095510387114, −7.37380764625589271157643448143, −6.70787258510936281557626899382, −5.47237984425161216057352953552, −5.14537644009540890269376378193, −4.89887786966406524902568686955, −3.59922092831737452605655479529, −3.19589212321753489661138271639, −2.29063387833110636461044791627, −1.14786333765116530667143396917,
1.14786333765116530667143396917, 2.29063387833110636461044791627, 3.19589212321753489661138271639, 3.59922092831737452605655479529, 4.89887786966406524902568686955, 5.14537644009540890269376378193, 5.47237984425161216057352953552, 6.70787258510936281557626899382, 7.37380764625589271157643448143, 7.898880420617599060095510387114
