L(s) = 1 | − 2-s + 3-s + 4-s + 0.532·5-s − 6-s + 1.87·7-s − 8-s + 9-s − 0.532·10-s + 11-s + 12-s − 1.34·13-s − 1.87·14-s + 0.532·15-s + 16-s − 2.06·17-s − 18-s − 6.10·19-s + 0.532·20-s + 1.87·21-s − 22-s − 2.34·23-s − 24-s − 4.71·25-s + 1.34·26-s + 27-s + 1.87·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.237·5-s − 0.408·6-s + 0.710·7-s − 0.353·8-s + 0.333·9-s − 0.168·10-s + 0.301·11-s + 0.288·12-s − 0.373·13-s − 0.502·14-s + 0.137·15-s + 0.250·16-s − 0.500·17-s − 0.235·18-s − 1.40·19-s + 0.118·20-s + 0.410·21-s − 0.213·22-s − 0.489·23-s − 0.204·24-s − 0.943·25-s + 0.264·26-s + 0.192·27-s + 0.355·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
good | 5 | \( 1 - 0.532T + 5T^{2} \) |
| 7 | \( 1 - 1.87T + 7T^{2} \) |
| 13 | \( 1 + 1.34T + 13T^{2} \) |
| 17 | \( 1 + 2.06T + 17T^{2} \) |
| 19 | \( 1 + 6.10T + 19T^{2} \) |
| 23 | \( 1 + 2.34T + 23T^{2} \) |
| 29 | \( 1 - 1.50T + 29T^{2} \) |
| 31 | \( 1 + 9.24T + 31T^{2} \) |
| 37 | \( 1 - 2.80T + 37T^{2} \) |
| 41 | \( 1 + 8.06T + 41T^{2} \) |
| 43 | \( 1 + 1.85T + 43T^{2} \) |
| 47 | \( 1 + 8.41T + 47T^{2} \) |
| 53 | \( 1 + 7.74T + 53T^{2} \) |
| 59 | \( 1 - 5.00T + 59T^{2} \) |
| 67 | \( 1 + 16.0T + 67T^{2} \) |
| 71 | \( 1 - 1.73T + 71T^{2} \) |
| 73 | \( 1 - 5.96T + 73T^{2} \) |
| 79 | \( 1 + 0.652T + 79T^{2} \) |
| 83 | \( 1 - 13.0T + 83T^{2} \) |
| 89 | \( 1 + 2.20T + 89T^{2} \) |
| 97 | \( 1 + 5.70T + 97T^{2} \) |
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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
−8.181825359023709804380101394864, −7.57489780240538482964784239351, −6.72810468015181401284014732446, −6.08367649435703144452674966383, −5.02034904932071385033118301015, −4.21626767593644847207337576296, −3.28376636387340296217272152613, −2.07072682259112377398279433552, −1.72285131930852739715669717683, 0,
1.72285131930852739715669717683, 2.07072682259112377398279433552, 3.28376636387340296217272152613, 4.21626767593644847207337576296, 5.02034904932071385033118301015, 6.08367649435703144452674966383, 6.72810468015181401284014732446, 7.57489780240538482964784239351, 8.181825359023709804380101394864