| L(s) = 1 | − 2-s + 3-s + 4-s + 4·5-s − 6-s − 8-s + 9-s − 4·10-s + 11-s + 12-s − 13-s + 4·15-s + 16-s + 17-s − 18-s + 2·19-s + 4·20-s − 22-s − 4·23-s − 24-s + 11·25-s + 26-s + 27-s + 2·29-s − 4·30-s − 32-s + 33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.78·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 1.26·10-s + 0.301·11-s + 0.288·12-s − 0.277·13-s + 1.03·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.458·19-s + 0.894·20-s − 0.213·22-s − 0.834·23-s − 0.204·24-s + 11/5·25-s + 0.196·26-s + 0.192·27-s + 0.371·29-s − 0.730·30-s − 0.176·32-s + 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.137035691\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.137035691\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−16.21090077914068, −15.69896300880333, −14.75114708819012, −14.39757895580699, −14.01024538916436, −13.34435805767725, −12.82206308061949, −12.25026707837201, −11.43831303817082, −10.81908638654190, −9.984590497350799, −9.815891489537064, −9.425168954332504, −8.625303306789774, −8.197513847943483, −7.398282044226276, −6.585007875943309, −6.338691273422551, −5.399185177651725, −4.968615566706756, −3.778117474162491, −3.007741232797420, −2.193082857089771, −1.776527385214073, −0.8578112312733917,
0.8578112312733917, 1.776527385214073, 2.193082857089771, 3.007741232797420, 3.778117474162491, 4.968615566706756, 5.399185177651725, 6.338691273422551, 6.585007875943309, 7.398282044226276, 8.197513847943483, 8.625303306789774, 9.425168954332504, 9.815891489537064, 9.984590497350799, 10.81908638654190, 11.43831303817082, 12.25026707837201, 12.82206308061949, 13.34435805767725, 14.01024538916436, 14.39757895580699, 14.75114708819012, 15.69896300880333, 16.21090077914068
