| L(s) = 1 | − 1.75·2-s − 1.67·3-s + 1.09·4-s + 3.67·5-s + 2.94·6-s − 4.14·7-s + 1.59·8-s − 0.197·9-s − 6.47·10-s + 5.07·11-s − 1.83·12-s + 7.30·14-s − 6.15·15-s − 4.99·16-s − 0.643·17-s + 0.347·18-s + 7.24·19-s + 4.03·20-s + 6.94·21-s − 8.93·22-s + 0.685·23-s − 2.66·24-s + 8.53·25-s + 5.35·27-s − 4.54·28-s + 5.82·29-s + 10.8·30-s + ⋯ |
| L(s) = 1 | − 1.24·2-s − 0.966·3-s + 0.547·4-s + 1.64·5-s + 1.20·6-s − 1.56·7-s + 0.562·8-s − 0.0657·9-s − 2.04·10-s + 1.53·11-s − 0.529·12-s + 1.95·14-s − 1.59·15-s − 1.24·16-s − 0.155·17-s + 0.0817·18-s + 1.66·19-s + 0.901·20-s + 1.51·21-s − 1.90·22-s + 0.142·23-s − 0.543·24-s + 1.70·25-s + 1.03·27-s − 0.859·28-s + 1.08·29-s + 1.97·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8568155087\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8568155087\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 1.75T + 2T^{2} \) |
| 3 | \( 1 + 1.67T + 3T^{2} \) |
| 5 | \( 1 - 3.67T + 5T^{2} \) |
| 7 | \( 1 + 4.14T + 7T^{2} \) |
| 11 | \( 1 - 5.07T + 11T^{2} \) |
| 17 | \( 1 + 0.643T + 17T^{2} \) |
| 19 | \( 1 - 7.24T + 19T^{2} \) |
| 23 | \( 1 - 0.685T + 23T^{2} \) |
| 29 | \( 1 - 5.82T + 29T^{2} \) |
| 37 | \( 1 + 8.72T + 37T^{2} \) |
| 41 | \( 1 - 11.1T + 41T^{2} \) |
| 43 | \( 1 + 1.81T + 43T^{2} \) |
| 47 | \( 1 + 6.05T + 47T^{2} \) |
| 53 | \( 1 + 1.56T + 53T^{2} \) |
| 59 | \( 1 - 7.14T + 59T^{2} \) |
| 61 | \( 1 - 1.54T + 61T^{2} \) |
| 67 | \( 1 + 2.10T + 67T^{2} \) |
| 71 | \( 1 + 0.546T + 71T^{2} \) |
| 73 | \( 1 - 9.58T + 73T^{2} \) |
| 79 | \( 1 + 4.15T + 79T^{2} \) |
| 83 | \( 1 + 6.77T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 - 12.2T + 97T^{2} \) |
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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
−8.586797704270984907092692475954, −7.27270268577452053640144566687, −6.60887741831247939968637219854, −6.31735358798921044900732634070, −5.57625088957268890070375081724, −4.80488530760125955632174314699, −3.52939889090137816565332933524, −2.57986008550809545778070645003, −1.39545593484006253041662928874, −0.71115430261891299297129372552,
0.71115430261891299297129372552, 1.39545593484006253041662928874, 2.57986008550809545778070645003, 3.52939889090137816565332933524, 4.80488530760125955632174314699, 5.57625088957268890070375081724, 6.31735358798921044900732634070, 6.60887741831247939968637219854, 7.27270268577452053640144566687, 8.586797704270984907092692475954
