| L(s) = 1 | + 15.9·2-s + 81·3-s − 256.·4-s + 61.0·5-s + 1.29e3·6-s + 4.75e3·7-s − 1.22e4·8-s + 6.56e3·9-s + 976.·10-s + 5.60e3·11-s − 2.07e4·12-s − 7.34e4·13-s + 7.60e4·14-s + 4.94e3·15-s − 6.49e4·16-s + 4.30e5·17-s + 1.04e5·18-s − 7.18e5·19-s − 1.56e4·20-s + 3.85e5·21-s + 8.95e4·22-s + 3.94e3·23-s − 9.94e5·24-s − 1.94e6·25-s − 1.17e6·26-s + 5.31e5·27-s − 1.22e6·28-s + ⋯ |
| L(s) = 1 | + 0.706·2-s + 0.577·3-s − 0.501·4-s + 0.0437·5-s + 0.407·6-s + 0.748·7-s − 1.06·8-s + 0.333·9-s + 0.0308·10-s + 0.115·11-s − 0.289·12-s − 0.713·13-s + 0.528·14-s + 0.0252·15-s − 0.247·16-s + 1.25·17-s + 0.235·18-s − 1.26·19-s − 0.0219·20-s + 0.432·21-s + 0.0815·22-s + 0.00293·23-s − 0.612·24-s − 0.998·25-s − 0.504·26-s + 0.192·27-s − 0.375·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 81T \) |
| 59 | \( 1 - 1.21e7T \) |
| good | 2 | \( 1 - 15.9T + 512T^{2} \) |
| 5 | \( 1 - 61.0T + 1.95e6T^{2} \) |
| 7 | \( 1 - 4.75e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 5.60e3T + 2.35e9T^{2} \) |
| 13 | \( 1 + 7.34e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 4.30e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 7.18e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 3.94e3T + 1.80e12T^{2} \) |
| 29 | \( 1 - 6.27e5T + 1.45e13T^{2} \) |
| 31 | \( 1 - 1.02e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.31e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 7.00e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.11e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 7.14e6T + 1.11e15T^{2} \) |
| 53 | \( 1 + 1.00e8T + 3.29e15T^{2} \) |
| 61 | \( 1 + 4.94e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.76e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.12e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.63e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 5.43e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 5.64e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 3.74e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.37e9T + 7.60e17T^{2} \) |
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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
−10.42995156718955564209807906832, −9.442618692062876935740685275326, −8.457817418582118659456516983570, −7.57832494297111148478085462486, −6.07292271037448532873155609639, −4.95856862870034185793564683247, −4.09013506190733766332922435443, −2.94534865491226763677713229812, −1.61629084179723735324273536812, 0,
1.61629084179723735324273536812, 2.94534865491226763677713229812, 4.09013506190733766332922435443, 4.95856862870034185793564683247, 6.07292271037448532873155609639, 7.57832494297111148478085462486, 8.457817418582118659456516983570, 9.442618692062876935740685275326, 10.42995156718955564209807906832
