| L(s) = 1 | − 1.10·2-s − 3.23·3-s − 0.787·4-s + 5-s + 3.56·6-s + 3.06·8-s + 7.49·9-s − 1.10·10-s − 2.98·11-s + 2.55·12-s + 1.14·13-s − 3.23·15-s − 1.80·16-s − 5.79·17-s − 8.24·18-s − 5.34·19-s − 0.787·20-s + 3.28·22-s − 3.18·23-s − 9.94·24-s + 25-s − 1.25·26-s − 14.5·27-s − 2.04·29-s + 3.56·30-s + 31-s − 4.15·32-s + ⋯ |
| L(s) = 1 | − 0.778·2-s − 1.87·3-s − 0.393·4-s + 0.447·5-s + 1.45·6-s + 1.08·8-s + 2.49·9-s − 0.348·10-s − 0.899·11-s + 0.736·12-s + 0.316·13-s − 0.836·15-s − 0.451·16-s − 1.40·17-s − 1.94·18-s − 1.22·19-s − 0.176·20-s + 0.700·22-s − 0.664·23-s − 2.02·24-s + 0.200·25-s − 0.246·26-s − 2.80·27-s − 0.379·29-s + 0.651·30-s + 0.179·31-s − 0.734·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7595 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7595 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1295619344\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1295619344\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 1.10T + 2T^{2} \) |
| 3 | \( 1 + 3.23T + 3T^{2} \) |
| 11 | \( 1 + 2.98T + 11T^{2} \) |
| 13 | \( 1 - 1.14T + 13T^{2} \) |
| 17 | \( 1 + 5.79T + 17T^{2} \) |
| 19 | \( 1 + 5.34T + 19T^{2} \) |
| 23 | \( 1 + 3.18T + 23T^{2} \) |
| 29 | \( 1 + 2.04T + 29T^{2} \) |
| 37 | \( 1 - 3.54T + 37T^{2} \) |
| 41 | \( 1 + 11.0T + 41T^{2} \) |
| 43 | \( 1 - 4.59T + 43T^{2} \) |
| 47 | \( 1 - 2.67T + 47T^{2} \) |
| 53 | \( 1 + 9.04T + 53T^{2} \) |
| 59 | \( 1 + 12.0T + 59T^{2} \) |
| 61 | \( 1 + 2.48T + 61T^{2} \) |
| 67 | \( 1 - 4.05T + 67T^{2} \) |
| 71 | \( 1 - 2.22T + 71T^{2} \) |
| 73 | \( 1 - 6.76T + 73T^{2} \) |
| 79 | \( 1 + 7.92T + 79T^{2} \) |
| 83 | \( 1 + 1.93T + 83T^{2} \) |
| 89 | \( 1 - 4.41T + 89T^{2} \) |
| 97 | \( 1 - 2.90T + 97T^{2} \) |
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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
−7.86444377695429166345661326351, −7.09986180114744451201756123572, −6.36407544411921080542640556097, −5.94461710130620547042586093364, −4.99698324723387291582724070875, −4.65728575902718377193227599641, −3.89268500744592059610387184883, −2.22947355484557439226529344963, −1.43224932544304200980154298221, −0.23673167699717175966620535357,
0.23673167699717175966620535357, 1.43224932544304200980154298221, 2.22947355484557439226529344963, 3.89268500744592059610387184883, 4.65728575902718377193227599641, 4.99698324723387291582724070875, 5.94461710130620547042586093364, 6.36407544411921080542640556097, 7.09986180114744451201756123572, 7.86444377695429166345661326351
