| L(s) = 1 | + 2-s + 3-s + 4-s − 2.34·5-s + 6-s + 4.92·7-s + 8-s + 9-s − 2.34·10-s + 0.147·11-s + 12-s − 3.88·13-s + 4.92·14-s − 2.34·15-s + 16-s + 7.05·17-s + 18-s + 19-s − 2.34·20-s + 4.92·21-s + 0.147·22-s + 2.33·23-s + 24-s + 0.489·25-s − 3.88·26-s + 27-s + 4.92·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.04·5-s + 0.408·6-s + 1.86·7-s + 0.353·8-s + 0.333·9-s − 0.740·10-s + 0.0444·11-s + 0.288·12-s − 1.07·13-s + 1.31·14-s − 0.604·15-s + 0.250·16-s + 1.71·17-s + 0.235·18-s + 0.229·19-s − 0.523·20-s + 1.07·21-s + 0.0314·22-s + 0.487·23-s + 0.204·24-s + 0.0979·25-s − 0.762·26-s + 0.192·27-s + 0.930·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.383360192\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.383360192\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 53 | \( 1 + T \) |
| good | 5 | \( 1 + 2.34T + 5T^{2} \) |
| 7 | \( 1 - 4.92T + 7T^{2} \) |
| 11 | \( 1 - 0.147T + 11T^{2} \) |
| 13 | \( 1 + 3.88T + 13T^{2} \) |
| 17 | \( 1 - 7.05T + 17T^{2} \) |
| 23 | \( 1 - 2.33T + 23T^{2} \) |
| 29 | \( 1 + 0.861T + 29T^{2} \) |
| 31 | \( 1 + 1.37T + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 - 4.12T + 41T^{2} \) |
| 43 | \( 1 + 9.14T + 43T^{2} \) |
| 47 | \( 1 + 12.6T + 47T^{2} \) |
| 59 | \( 1 - 11.6T + 59T^{2} \) |
| 61 | \( 1 - 1.78T + 61T^{2} \) |
| 67 | \( 1 - 0.00540T + 67T^{2} \) |
| 71 | \( 1 - 6.38T + 71T^{2} \) |
| 73 | \( 1 - 8.90T + 73T^{2} \) |
| 79 | \( 1 - 3.67T + 79T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 + 13.9T + 89T^{2} \) |
| 97 | \( 1 - 11.9T + 97T^{2} \) |
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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
−8.084825190185630221186440483409, −7.52108710729979420832231982494, −6.91880432224830031922388138941, −5.61761159329347200635041373448, −5.03052009357365477741104724738, −4.46046919733833136575931031735, −3.73245975280875394586009076077, −2.92856692048083817739737391963, −1.98230486574382847105663398831, −1.02692830127875923300312132109,
1.02692830127875923300312132109, 1.98230486574382847105663398831, 2.92856692048083817739737391963, 3.73245975280875394586009076077, 4.46046919733833136575931031735, 5.03052009357365477741104724738, 5.61761159329347200635041373448, 6.91880432224830031922388138941, 7.52108710729979420832231982494, 8.084825190185630221186440483409
