L(s) = 1 | + 1.79·3-s + 0.876·5-s + 7-s + 0.233·9-s − 11-s + 13-s + 1.57·15-s − 1.98·17-s − 7.48·19-s + 1.79·21-s − 5.93·23-s − 4.23·25-s − 4.97·27-s + 4.29·29-s + 3.81·31-s − 1.79·33-s + 0.876·35-s − 4.36·37-s + 1.79·39-s − 4.55·41-s + 12.2·43-s + 0.204·45-s + 8.41·47-s + 49-s − 3.56·51-s + 1.66·53-s − 0.876·55-s + ⋯ |
L(s) = 1 | + 1.03·3-s + 0.391·5-s + 0.377·7-s + 0.0776·9-s − 0.301·11-s + 0.277·13-s + 0.406·15-s − 0.480·17-s − 1.71·19-s + 0.392·21-s − 1.23·23-s − 0.846·25-s − 0.957·27-s + 0.797·29-s + 0.685·31-s − 0.313·33-s + 0.148·35-s − 0.718·37-s + 0.287·39-s − 0.711·41-s + 1.86·43-s + 0.0304·45-s + 1.22·47-s + 0.142·49-s − 0.499·51-s + 0.228·53-s − 0.118·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 3 | \( 1 - 1.79T + 3T^{2} \) |
| 5 | \( 1 - 0.876T + 5T^{2} \) |
| 17 | \( 1 + 1.98T + 17T^{2} \) |
| 19 | \( 1 + 7.48T + 19T^{2} \) |
| 23 | \( 1 + 5.93T + 23T^{2} \) |
| 29 | \( 1 - 4.29T + 29T^{2} \) |
| 31 | \( 1 - 3.81T + 31T^{2} \) |
| 37 | \( 1 + 4.36T + 37T^{2} \) |
| 41 | \( 1 + 4.55T + 41T^{2} \) |
| 43 | \( 1 - 12.2T + 43T^{2} \) |
| 47 | \( 1 - 8.41T + 47T^{2} \) |
| 53 | \( 1 - 1.66T + 53T^{2} \) |
| 59 | \( 1 + 8.44T + 59T^{2} \) |
| 61 | \( 1 + 4.74T + 61T^{2} \) |
| 67 | \( 1 + 1.03T + 67T^{2} \) |
| 71 | \( 1 - 6.77T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 + 4.82T + 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 - 12.0T + 97T^{2} \) |
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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
−7.77238758342398580758827097504, −6.82217339102436900750669001757, −6.07281700351120935357504264823, −5.53875556797544430504614145762, −4.31026510035181651585103603380, −4.07126257939552852277801591111, −2.87287712171161656681100311730, −2.31494222719130867085654478908, −1.61014566850716425107217072364, 0,
1.61014566850716425107217072364, 2.31494222719130867085654478908, 2.87287712171161656681100311730, 4.07126257939552852277801591111, 4.31026510035181651585103603380, 5.53875556797544430504614145762, 6.07281700351120935357504264823, 6.82217339102436900750669001757, 7.77238758342398580758827097504