| L(s) = 1 | + 1.93·3-s − 5-s + 4.93·7-s + 0.745·9-s − 0.745·11-s − 1.74·13-s − 1.93·15-s − 6.10·17-s − 5.44·19-s + 9.55·21-s − 23-s + 25-s − 4.36·27-s + 1.66·29-s − 1.61·31-s − 1.44·33-s − 4.93·35-s − 4.34·37-s − 3.37·39-s − 6.95·41-s − 5.01·43-s − 0.745·45-s + 2.68·47-s + 17.3·49-s − 11.8·51-s − 13.7·53-s + 0.745·55-s + ⋯ |
| L(s) = 1 | + 1.11·3-s − 0.447·5-s + 1.86·7-s + 0.248·9-s − 0.224·11-s − 0.484·13-s − 0.499·15-s − 1.48·17-s − 1.24·19-s + 2.08·21-s − 0.208·23-s + 0.200·25-s − 0.839·27-s + 0.309·29-s − 0.290·31-s − 0.251·33-s − 0.834·35-s − 0.714·37-s − 0.541·39-s − 1.08·41-s − 0.764·43-s − 0.111·45-s + 0.391·47-s + 2.47·49-s − 1.65·51-s − 1.88·53-s + 0.100·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 1.93T + 3T^{2} \) |
| 7 | \( 1 - 4.93T + 7T^{2} \) |
| 11 | \( 1 + 0.745T + 11T^{2} \) |
| 13 | \( 1 + 1.74T + 13T^{2} \) |
| 17 | \( 1 + 6.10T + 17T^{2} \) |
| 19 | \( 1 + 5.44T + 19T^{2} \) |
| 29 | \( 1 - 1.66T + 29T^{2} \) |
| 31 | \( 1 + 1.61T + 31T^{2} \) |
| 37 | \( 1 + 4.34T + 37T^{2} \) |
| 41 | \( 1 + 6.95T + 41T^{2} \) |
| 43 | \( 1 + 5.01T + 43T^{2} \) |
| 47 | \( 1 - 2.68T + 47T^{2} \) |
| 53 | \( 1 + 13.7T + 53T^{2} \) |
| 59 | \( 1 + 12.2T + 59T^{2} \) |
| 61 | \( 1 - 13.9T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 + 9.67T + 71T^{2} \) |
| 73 | \( 1 - 5.69T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 + 0.637T + 83T^{2} \) |
| 89 | \( 1 - 2.72T + 89T^{2} \) |
| 97 | \( 1 - 7.12T + 97T^{2} \) |
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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
−7.77765166756282836263002718886, −7.12697445079586952020779925679, −6.27440867481884422080990482039, −5.15179909896087799003100493505, −4.61711297626532324867930910286, −4.04156069406578933628724374454, −3.04268594789299868524256124234, −2.12254244134843212810943381046, −1.73471375913162315925989845679, 0,
1.73471375913162315925989845679, 2.12254244134843212810943381046, 3.04268594789299868524256124234, 4.04156069406578933628724374454, 4.61711297626532324867930910286, 5.15179909896087799003100493505, 6.27440867481884422080990482039, 7.12697445079586952020779925679, 7.77765166756282836263002718886
