| L(s) = 1 | − 1.96·2-s − 0.967·3-s + 1.87·4-s + 0.188·5-s + 1.90·6-s − 4.91·7-s + 0.250·8-s − 2.06·9-s − 0.371·10-s − 1.60·11-s − 1.81·12-s + 3.63·13-s + 9.67·14-s − 0.182·15-s − 4.23·16-s + 3.34·17-s + 4.06·18-s − 19-s + 0.353·20-s + 4.75·21-s + 3.16·22-s + 2.14·23-s − 0.242·24-s − 4.96·25-s − 7.15·26-s + 4.89·27-s − 9.20·28-s + ⋯ |
| L(s) = 1 | − 1.39·2-s − 0.558·3-s + 0.936·4-s + 0.0844·5-s + 0.777·6-s − 1.85·7-s + 0.0884·8-s − 0.687·9-s − 0.117·10-s − 0.485·11-s − 0.523·12-s + 1.00·13-s + 2.58·14-s − 0.0471·15-s − 1.05·16-s + 0.811·17-s + 0.957·18-s − 0.229·19-s + 0.0790·20-s + 1.03·21-s + 0.674·22-s + 0.447·23-s − 0.0494·24-s − 0.992·25-s − 1.40·26-s + 0.942·27-s − 1.73·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 19 | \( 1 + T \) |
| 317 | \( 1 + T \) |
| good | 2 | \( 1 + 1.96T + 2T^{2} \) |
| 3 | \( 1 + 0.967T + 3T^{2} \) |
| 5 | \( 1 - 0.188T + 5T^{2} \) |
| 7 | \( 1 + 4.91T + 7T^{2} \) |
| 11 | \( 1 + 1.60T + 11T^{2} \) |
| 13 | \( 1 - 3.63T + 13T^{2} \) |
| 17 | \( 1 - 3.34T + 17T^{2} \) |
| 23 | \( 1 - 2.14T + 23T^{2} \) |
| 29 | \( 1 + 3.45T + 29T^{2} \) |
| 31 | \( 1 - 3.44T + 31T^{2} \) |
| 37 | \( 1 + 5.13T + 37T^{2} \) |
| 41 | \( 1 + 11.7T + 41T^{2} \) |
| 43 | \( 1 + 3.74T + 43T^{2} \) |
| 47 | \( 1 - 5.82T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 - 5.17T + 59T^{2} \) |
| 61 | \( 1 - 9.16T + 61T^{2} \) |
| 67 | \( 1 - 13.8T + 67T^{2} \) |
| 71 | \( 1 + 7.81T + 71T^{2} \) |
| 73 | \( 1 - 7.15T + 73T^{2} \) |
| 79 | \( 1 - 14.7T + 79T^{2} \) |
| 83 | \( 1 - 8.05T + 83T^{2} \) |
| 89 | \( 1 - 0.0958T + 89T^{2} \) |
| 97 | \( 1 + 3.34T + 97T^{2} \) |
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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
−7.962149492176225261506698624540, −6.92461760494651115525776192111, −6.53616146082609975449075453387, −5.81080090257177977770111263107, −5.12430098346178104113285474129, −3.73627889472055619750200294333, −3.16821044065014040841458962846, −2.09318212659065779361098779614, −0.809003634664333666865494214724, 0,
0.809003634664333666865494214724, 2.09318212659065779361098779614, 3.16821044065014040841458962846, 3.73627889472055619750200294333, 5.12430098346178104113285474129, 5.81080090257177977770111263107, 6.53616146082609975449075453387, 6.92461760494651115525776192111, 7.962149492176225261506698624540
