L(s) = 1 | + 28.6·2-s + 141.·3-s − 1.22e3·4-s − 2.64e3·5-s + 4.04e3·6-s + 7.64e4·7-s − 9.37e4·8-s − 1.57e5·9-s − 7.57e4·10-s − 6.03e5·11-s − 1.73e5·12-s − 9.41e5·13-s + 2.18e6·14-s − 3.74e5·15-s − 1.65e5·16-s − 5.54e6·17-s − 4.49e6·18-s − 8.15e6·19-s + 3.25e6·20-s + 1.08e7·21-s − 1.72e7·22-s + 6.43e6·23-s − 1.32e7·24-s − 4.18e7·25-s − 2.69e7·26-s − 4.72e7·27-s − 9.39e7·28-s + ⋯ |
L(s) = 1 | + 0.632·2-s + 0.336·3-s − 0.600·4-s − 0.378·5-s + 0.212·6-s + 1.71·7-s − 1.01·8-s − 0.887·9-s − 0.239·10-s − 1.12·11-s − 0.201·12-s − 0.703·13-s + 1.08·14-s − 0.127·15-s − 0.0394·16-s − 0.947·17-s − 0.560·18-s − 0.755·19-s + 0.227·20-s + 0.577·21-s − 0.714·22-s + 0.208·23-s − 0.340·24-s − 0.856·25-s − 0.444·26-s − 0.634·27-s − 1.03·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 23 | \( 1 - 6.43e6T \) |
good | 2 | \( 1 - 28.6T + 2.04e3T^{2} \) |
| 3 | \( 1 - 141.T + 1.77e5T^{2} \) |
| 5 | \( 1 + 2.64e3T + 4.88e7T^{2} \) |
| 7 | \( 1 - 7.64e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 6.03e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 9.41e5T + 1.79e12T^{2} \) |
| 17 | \( 1 + 5.54e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 8.15e6T + 1.16e14T^{2} \) |
| 29 | \( 1 - 1.14e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 1.09e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 3.97e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 7.31e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 7.98e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 4.30e8T + 2.47e18T^{2} \) |
| 53 | \( 1 - 4.54e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 7.83e8T + 3.01e19T^{2} \) |
| 61 | \( 1 + 7.23e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 1.06e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + 1.08e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 3.25e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 5.15e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 6.27e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 2.11e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 1.36e11T + 7.15e21T^{2} \) |
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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
−14.52076073145702315096612398040, −13.57002958266378082813427963259, −12.10456306479385019411504751416, −10.85900652027330498347477049662, −8.799997257620092501523482552146, −7.87321887208697347504487800387, −5.41065272263762032386932266019, −4.36366449123772299926770985435, −2.44957969283203089533100686772, 0,
2.44957969283203089533100686772, 4.36366449123772299926770985435, 5.41065272263762032386932266019, 7.87321887208697347504487800387, 8.799997257620092501523482552146, 10.85900652027330498347477049662, 12.10456306479385019411504751416, 13.57002958266378082813427963259, 14.52076073145702315096612398040