L(s) = 1 | + 2-s + 3-s + 4-s + 1.64·5-s + 6-s + 3.14·7-s + 8-s + 9-s + 1.64·10-s − 0.445·11-s + 12-s − 13-s + 3.14·14-s + 1.64·15-s + 16-s + 2.45·17-s + 18-s + 5.11·19-s + 1.64·20-s + 3.14·21-s − 0.445·22-s + 5.71·23-s + 24-s − 2.28·25-s − 26-s + 27-s + 3.14·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.736·5-s + 0.408·6-s + 1.18·7-s + 0.353·8-s + 0.333·9-s + 0.520·10-s − 0.134·11-s + 0.288·12-s − 0.277·13-s + 0.840·14-s + 0.425·15-s + 0.250·16-s + 0.595·17-s + 0.235·18-s + 1.17·19-s + 0.368·20-s + 0.685·21-s − 0.0949·22-s + 1.19·23-s + 0.204·24-s − 0.457·25-s − 0.196·26-s + 0.192·27-s + 0.594·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.142810847\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.142810847\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
good | 5 | \( 1 - 1.64T + 5T^{2} \) |
| 7 | \( 1 - 3.14T + 7T^{2} \) |
| 11 | \( 1 + 0.445T + 11T^{2} \) |
| 17 | \( 1 - 2.45T + 17T^{2} \) |
| 19 | \( 1 - 5.11T + 19T^{2} \) |
| 23 | \( 1 - 5.71T + 23T^{2} \) |
| 29 | \( 1 + 0.516T + 29T^{2} \) |
| 31 | \( 1 - 4.19T + 31T^{2} \) |
| 37 | \( 1 + 4.57T + 37T^{2} \) |
| 41 | \( 1 + 8.86T + 41T^{2} \) |
| 43 | \( 1 + 0.324T + 43T^{2} \) |
| 47 | \( 1 + 1.57T + 47T^{2} \) |
| 53 | \( 1 - 3.81T + 53T^{2} \) |
| 59 | \( 1 - 3.45T + 59T^{2} \) |
| 61 | \( 1 + 0.127T + 61T^{2} \) |
| 67 | \( 1 - 8.34T + 67T^{2} \) |
| 71 | \( 1 + 13.1T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 + 12.2T + 79T^{2} \) |
| 83 | \( 1 - 8.54T + 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 - 4.87T + 97T^{2} \) |
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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
−7.70955884407386959616391502899, −7.21928494764901299018576946165, −6.42164301247405682460765252024, −5.38384617105837840854363335618, −5.19937016746700401153460928239, −4.38157083303641284934626054482, −3.41516271406055136626790955054, −2.76006909076246412602355313057, −1.85209759797776693054592715548, −1.20767329727359477131783546416,
1.20767329727359477131783546416, 1.85209759797776693054592715548, 2.76006909076246412602355313057, 3.41516271406055136626790955054, 4.38157083303641284934626054482, 5.19937016746700401153460928239, 5.38384617105837840854363335618, 6.42164301247405682460765252024, 7.21928494764901299018576946165, 7.70955884407386959616391502899