L(s) = 1 | − 2-s + 1.34·3-s + 4-s − 1.34·6-s + 0.833·7-s − 8-s − 1.18·9-s + 0.833·11-s + 1.34·12-s − 4.56·13-s − 0.833·14-s + 16-s − 5.45·17-s + 1.18·18-s − 8.65·19-s + 1.12·21-s − 0.833·22-s − 3.53·23-s − 1.34·24-s + 4.56·26-s − 5.63·27-s + 0.833·28-s + 3.26·29-s + 6.00·31-s − 32-s + 1.12·33-s + 5.45·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.778·3-s + 0.5·4-s − 0.550·6-s + 0.314·7-s − 0.353·8-s − 0.393·9-s + 0.251·11-s + 0.389·12-s − 1.26·13-s − 0.222·14-s + 0.250·16-s − 1.32·17-s + 0.278·18-s − 1.98·19-s + 0.245·21-s − 0.177·22-s − 0.736·23-s − 0.275·24-s + 0.895·26-s − 1.08·27-s + 0.157·28-s + 0.607·29-s + 1.07·31-s − 0.176·32-s + 0.195·33-s + 0.934·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - 1.34T + 3T^{2} \) |
| 7 | \( 1 - 0.833T + 7T^{2} \) |
| 11 | \( 1 - 0.833T + 11T^{2} \) |
| 13 | \( 1 + 4.56T + 13T^{2} \) |
| 17 | \( 1 + 5.45T + 17T^{2} \) |
| 19 | \( 1 + 8.65T + 19T^{2} \) |
| 23 | \( 1 + 3.53T + 23T^{2} \) |
| 29 | \( 1 - 3.26T + 29T^{2} \) |
| 31 | \( 1 - 6.00T + 31T^{2} \) |
| 37 | \( 1 + 7.31T + 37T^{2} \) |
| 41 | \( 1 - 1.86T + 41T^{2} \) |
| 43 | \( 1 - 1.63T + 43T^{2} \) |
| 47 | \( 1 - 0.833T + 47T^{2} \) |
| 53 | \( 1 + 6.42T + 53T^{2} \) |
| 59 | \( 1 - 13.5T + 59T^{2} \) |
| 61 | \( 1 - 5.88T + 61T^{2} \) |
| 67 | \( 1 + 1.49T + 67T^{2} \) |
| 71 | \( 1 + 2.36T + 71T^{2} \) |
| 73 | \( 1 - 3.33T + 73T^{2} \) |
| 79 | \( 1 - 5.16T + 79T^{2} \) |
| 83 | \( 1 - 12.3T + 83T^{2} \) |
| 89 | \( 1 + 7.00T + 89T^{2} \) |
| 97 | \( 1 + 11.0T + 97T^{2} \) |
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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
−9.130096411825217355301891974929, −8.464864608077114469503591835066, −8.016536180639195454710961108843, −6.90992593567912865501736429294, −6.26445338018639284302692759243, −4.92165365354045131767745289165, −3.97413212282860475033585764236, −2.58270644129327509562017968579, −2.04283183739455765973719126610, 0,
2.04283183739455765973719126610, 2.58270644129327509562017968579, 3.97413212282860475033585764236, 4.92165365354045131767745289165, 6.26445338018639284302692759243, 6.90992593567912865501736429294, 8.016536180639195454710961108843, 8.464864608077114469503591835066, 9.130096411825217355301891974929