| L(s) = 1 | + 32·2-s − 90·3-s + 1.02e3·4-s − 7.48e3·5-s − 2.88e3·6-s − 1.68e4·7-s + 3.27e4·8-s − 1.69e5·9-s − 2.39e5·10-s − 2.94e5·11-s − 9.21e4·12-s − 2.10e5·13-s − 5.37e5·14-s + 6.73e5·15-s + 1.04e6·16-s − 6.96e6·17-s − 5.40e6·18-s − 9.34e6·19-s − 7.65e6·20-s + 1.51e6·21-s − 9.42e6·22-s + 5.11e7·23-s − 2.94e6·24-s + 7.12e6·25-s − 6.73e6·26-s + 3.11e7·27-s − 1.72e7·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.213·3-s + 1/2·4-s − 1.07·5-s − 0.151·6-s − 0.377·7-s + 0.353·8-s − 0.954·9-s − 0.756·10-s − 0.551·11-s − 0.106·12-s − 0.157·13-s − 0.267·14-s + 0.228·15-s + 1/4·16-s − 1.18·17-s − 0.674·18-s − 0.865·19-s − 0.535·20-s + 0.0808·21-s − 0.389·22-s + 1.65·23-s − 0.0756·24-s + 0.145·25-s − 0.111·26-s + 0.417·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - p^{5} T \) |
| 7 | \( 1 + p^{5} T \) |
| good | 3 | \( 1 + 10 p^{2} T + p^{11} T^{2} \) |
| 5 | \( 1 + 1496 p T + p^{11} T^{2} \) |
| 11 | \( 1 + 26776 p T + p^{11} T^{2} \) |
| 13 | \( 1 + 210588 T + p^{11} T^{2} \) |
| 17 | \( 1 + 6962906 T + p^{11} T^{2} \) |
| 19 | \( 1 + 9346390 T + p^{11} T^{2} \) |
| 23 | \( 1 - 51172000 T + p^{11} T^{2} \) |
| 29 | \( 1 - 166196354 T + p^{11} T^{2} \) |
| 31 | \( 1 - 119000988 T + p^{11} T^{2} \) |
| 37 | \( 1 + 275545510 T + p^{11} T^{2} \) |
| 41 | \( 1 + 197988378 T + p^{11} T^{2} \) |
| 43 | \( 1 + 809489728 T + p^{11} T^{2} \) |
| 47 | \( 1 + 2600196204 T + p^{11} T^{2} \) |
| 53 | \( 1 - 733631454 T + p^{11} T^{2} \) |
| 59 | \( 1 + 4657126942 T + p^{11} T^{2} \) |
| 61 | \( 1 + 5135837424 T + p^{11} T^{2} \) |
| 67 | \( 1 - 8810564836 T + p^{11} T^{2} \) |
| 71 | \( 1 + 3849006656 T + p^{11} T^{2} \) |
| 73 | \( 1 + 18686748254 T + p^{11} T^{2} \) |
| 79 | \( 1 + 29850061992 T + p^{11} T^{2} \) |
| 83 | \( 1 + 5875980446 T + p^{11} T^{2} \) |
| 89 | \( 1 - 83056539450 T + p^{11} T^{2} \) |
| 97 | \( 1 - 149400800374 T + p^{11} T^{2} \) |
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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
−15.91750776244037102862446487073, −14.94467009204333427145328319498, −13.30546474715620910109072691513, −11.94618254706632678082150667087, −10.82981125192731860924711273649, −8.425982558695146719194879290614, −6.64462797642940359138451685488, −4.74361923187601662194011739430, −2.96621046873162550438699257196, 0,
2.96621046873162550438699257196, 4.74361923187601662194011739430, 6.64462797642940359138451685488, 8.425982558695146719194879290614, 10.82981125192731860924711273649, 11.94618254706632678082150667087, 13.30546474715620910109072691513, 14.94467009204333427145328319498, 15.91750776244037102862446487073
