| L(s) = 1 | − 2-s − 1.39·3-s + 4-s + 3.35·5-s + 1.39·6-s − 0.849·7-s − 8-s − 1.06·9-s − 3.35·10-s − 2.14·11-s − 1.39·12-s − 0.777·13-s + 0.849·14-s − 4.67·15-s + 16-s + 4.54·17-s + 1.06·18-s − 19-s + 3.35·20-s + 1.18·21-s + 2.14·22-s − 6.97·23-s + 1.39·24-s + 6.27·25-s + 0.777·26-s + 5.65·27-s − 0.849·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.803·3-s + 0.5·4-s + 1.50·5-s + 0.568·6-s − 0.321·7-s − 0.353·8-s − 0.353·9-s − 1.06·10-s − 0.646·11-s − 0.401·12-s − 0.215·13-s + 0.227·14-s − 1.20·15-s + 0.250·16-s + 1.10·17-s + 0.250·18-s − 0.229·19-s + 0.750·20-s + 0.258·21-s + 0.457·22-s − 1.45·23-s + 0.284·24-s + 1.25·25-s + 0.152·26-s + 1.08·27-s − 0.160·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 + 1.39T + 3T^{2} \) |
| 5 | \( 1 - 3.35T + 5T^{2} \) |
| 7 | \( 1 + 0.849T + 7T^{2} \) |
| 11 | \( 1 + 2.14T + 11T^{2} \) |
| 13 | \( 1 + 0.777T + 13T^{2} \) |
| 17 | \( 1 - 4.54T + 17T^{2} \) |
| 23 | \( 1 + 6.97T + 23T^{2} \) |
| 29 | \( 1 + 0.251T + 29T^{2} \) |
| 31 | \( 1 + 8.19T + 31T^{2} \) |
| 37 | \( 1 - 7.04T + 37T^{2} \) |
| 41 | \( 1 - 3.52T + 41T^{2} \) |
| 43 | \( 1 - 7.32T + 43T^{2} \) |
| 47 | \( 1 + 4.32T + 47T^{2} \) |
| 53 | \( 1 - 11.9T + 53T^{2} \) |
| 59 | \( 1 + 3.78T + 59T^{2} \) |
| 61 | \( 1 - 7.33T + 61T^{2} \) |
| 67 | \( 1 + 6.16T + 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 + 3.21T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 - 3.27T + 89T^{2} \) |
| 97 | \( 1 + 7.35T + 97T^{2} \) |
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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
−7.53189595597461437096843081016, −6.64018413953080161416037416917, −5.98955500557488174591384441824, −5.65752452209887466664791946954, −5.09423167380090548367184163173, −3.84205990245348609668688398641, −2.72272219845041916872107778115, −2.17018501210388722075525343577, −1.13425936938956169697005354295, 0,
1.13425936938956169697005354295, 2.17018501210388722075525343577, 2.72272219845041916872107778115, 3.84205990245348609668688398641, 5.09423167380090548367184163173, 5.65752452209887466664791946954, 5.98955500557488174591384441824, 6.64018413953080161416037416917, 7.53189595597461437096843081016
