| L(s) = 1 | + 1.78·2-s + 3-s + 1.19·4-s − 5-s + 1.78·6-s + 3.13·7-s − 1.43·8-s + 9-s − 1.78·10-s − 1.61·11-s + 1.19·12-s − 0.857·13-s + 5.60·14-s − 15-s − 4.96·16-s − 4.56·17-s + 1.78·18-s − 6.37·19-s − 1.19·20-s + 3.13·21-s − 2.88·22-s − 3.38·23-s − 1.43·24-s + 25-s − 1.53·26-s + 27-s + 3.75·28-s + ⋯ |
| L(s) = 1 | + 1.26·2-s + 0.577·3-s + 0.598·4-s − 0.447·5-s + 0.729·6-s + 1.18·7-s − 0.507·8-s + 0.333·9-s − 0.565·10-s − 0.486·11-s + 0.345·12-s − 0.237·13-s + 1.49·14-s − 0.258·15-s − 1.24·16-s − 1.10·17-s + 0.421·18-s − 1.46·19-s − 0.267·20-s + 0.684·21-s − 0.614·22-s − 0.706·23-s − 0.293·24-s + 0.200·25-s − 0.300·26-s + 0.192·27-s + 0.709·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 - 1.78T + 2T^{2} \) |
| 7 | \( 1 - 3.13T + 7T^{2} \) |
| 11 | \( 1 + 1.61T + 11T^{2} \) |
| 13 | \( 1 + 0.857T + 13T^{2} \) |
| 17 | \( 1 + 4.56T + 17T^{2} \) |
| 19 | \( 1 + 6.37T + 19T^{2} \) |
| 23 | \( 1 + 3.38T + 23T^{2} \) |
| 29 | \( 1 - 5.37T + 29T^{2} \) |
| 31 | \( 1 + 2.52T + 31T^{2} \) |
| 37 | \( 1 + 6.21T + 37T^{2} \) |
| 41 | \( 1 - 7.61T + 41T^{2} \) |
| 43 | \( 1 + 6.36T + 43T^{2} \) |
| 47 | \( 1 + 13.0T + 47T^{2} \) |
| 53 | \( 1 + 2.17T + 53T^{2} \) |
| 59 | \( 1 - 3.21T + 59T^{2} \) |
| 61 | \( 1 + 3.23T + 61T^{2} \) |
| 67 | \( 1 - 6.93T + 67T^{2} \) |
| 71 | \( 1 + 0.556T + 71T^{2} \) |
| 73 | \( 1 + 10.3T + 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 + 1.17T + 83T^{2} \) |
| 89 | \( 1 - 10.2T + 89T^{2} \) |
| 97 | \( 1 + 8.69T + 97T^{2} \) |
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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
−7.80635617045703598618523407293, −6.82935340712535189575477946914, −6.29235815312723690682648385160, −5.23446737718227649392515853599, −4.66338964738543171590841857353, −4.23945109932082911666035570059, −3.41295775728920834928603915801, −2.46398305215540609278919151153, −1.83373890795629242771549201640, 0,
1.83373890795629242771549201640, 2.46398305215540609278919151153, 3.41295775728920834928603915801, 4.23945109932082911666035570059, 4.66338964738543171590841857353, 5.23446737718227649392515853599, 6.29235815312723690682648385160, 6.82935340712535189575477946914, 7.80635617045703598618523407293
