L(s) = 1 | + 2-s − 1.82·3-s + 4-s − 5-s − 1.82·6-s + 3.00·7-s + 8-s + 0.335·9-s − 10-s − 11-s − 1.82·12-s − 4.32·13-s + 3.00·14-s + 1.82·15-s + 16-s − 2.32·17-s + 0.335·18-s − 1.98·19-s − 20-s − 5.48·21-s − 22-s + 6.29·23-s − 1.82·24-s + 25-s − 4.32·26-s + 4.86·27-s + 3.00·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.05·3-s + 0.5·4-s − 0.447·5-s − 0.745·6-s + 1.13·7-s + 0.353·8-s + 0.111·9-s − 0.316·10-s − 0.301·11-s − 0.527·12-s − 1.19·13-s + 0.802·14-s + 0.471·15-s + 0.250·16-s − 0.565·17-s + 0.0791·18-s − 0.454·19-s − 0.223·20-s − 1.19·21-s − 0.213·22-s + 1.31·23-s − 0.372·24-s + 0.200·25-s − 0.848·26-s + 0.936·27-s + 0.567·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 - T \) |
good | 3 | \( 1 + 1.82T + 3T^{2} \) |
| 7 | \( 1 - 3.00T + 7T^{2} \) |
| 13 | \( 1 + 4.32T + 13T^{2} \) |
| 17 | \( 1 + 2.32T + 17T^{2} \) |
| 19 | \( 1 + 1.98T + 19T^{2} \) |
| 23 | \( 1 - 6.29T + 23T^{2} \) |
| 29 | \( 1 - 3.78T + 29T^{2} \) |
| 31 | \( 1 - 2.81T + 31T^{2} \) |
| 37 | \( 1 + 0.856T + 37T^{2} \) |
| 41 | \( 1 + 5.60T + 41T^{2} \) |
| 43 | \( 1 - 8.01T + 43T^{2} \) |
| 47 | \( 1 + 6.59T + 47T^{2} \) |
| 53 | \( 1 + 9.34T + 53T^{2} \) |
| 59 | \( 1 - 7.84T + 59T^{2} \) |
| 61 | \( 1 + 2.89T + 61T^{2} \) |
| 67 | \( 1 + 9.07T + 67T^{2} \) |
| 71 | \( 1 - 4.45T + 71T^{2} \) |
| 79 | \( 1 - 5.72T + 79T^{2} \) |
| 83 | \( 1 + 5.58T + 83T^{2} \) |
| 89 | \( 1 + 6.01T + 89T^{2} \) |
| 97 | \( 1 + 7.84T + 97T^{2} \) |
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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
−7.24752662215222304118943152181, −6.75675987948321455957899491523, −5.98845196558182427097148005112, −5.07594619713815166466460579099, −4.88658979496133879305159339081, −4.30497713180684647978611868625, −3.08632395654248289470448025631, −2.35096795936989614648694963527, −1.22223470276789757342223816014, 0,
1.22223470276789757342223816014, 2.35096795936989614648694963527, 3.08632395654248289470448025631, 4.30497713180684647978611868625, 4.88658979496133879305159339081, 5.07594619713815166466460579099, 5.98845196558182427097148005112, 6.75675987948321455957899491523, 7.24752662215222304118943152181