| L(s) = 1 | − 1.27·2-s + 0.253·3-s − 0.378·4-s + 2.03·5-s − 0.323·6-s − 2.19·7-s + 3.02·8-s − 2.93·9-s − 2.59·10-s − 11-s − 0.0961·12-s + 3.83·13-s + 2.78·14-s + 0.517·15-s − 3.09·16-s + 3.73·18-s − 3.47·19-s − 0.772·20-s − 0.556·21-s + 1.27·22-s + 2.80·23-s + 0.769·24-s − 0.840·25-s − 4.88·26-s − 1.50·27-s + 0.829·28-s + 5.54·29-s + ⋯ |
| L(s) = 1 | − 0.900·2-s + 0.146·3-s − 0.189·4-s + 0.912·5-s − 0.131·6-s − 0.828·7-s + 1.07·8-s − 0.978·9-s − 0.821·10-s − 0.301·11-s − 0.0277·12-s + 1.06·13-s + 0.745·14-s + 0.133·15-s − 0.774·16-s + 0.881·18-s − 0.798·19-s − 0.172·20-s − 0.121·21-s + 0.271·22-s + 0.583·23-s + 0.156·24-s − 0.168·25-s − 0.958·26-s − 0.290·27-s + 0.156·28-s + 1.02·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3179 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3179 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9208166788\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9208166788\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 1.27T + 2T^{2} \) |
| 3 | \( 1 - 0.253T + 3T^{2} \) |
| 5 | \( 1 - 2.03T + 5T^{2} \) |
| 7 | \( 1 + 2.19T + 7T^{2} \) |
| 13 | \( 1 - 3.83T + 13T^{2} \) |
| 19 | \( 1 + 3.47T + 19T^{2} \) |
| 23 | \( 1 - 2.80T + 23T^{2} \) |
| 29 | \( 1 - 5.54T + 29T^{2} \) |
| 31 | \( 1 + 2.96T + 31T^{2} \) |
| 37 | \( 1 - 5.80T + 37T^{2} \) |
| 41 | \( 1 + 8.84T + 41T^{2} \) |
| 43 | \( 1 - 3.46T + 43T^{2} \) |
| 47 | \( 1 - 6.12T + 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 - 4.37T + 59T^{2} \) |
| 61 | \( 1 + 6.67T + 61T^{2} \) |
| 67 | \( 1 + 14.1T + 67T^{2} \) |
| 71 | \( 1 + 8.96T + 71T^{2} \) |
| 73 | \( 1 - 0.981T + 73T^{2} \) |
| 79 | \( 1 - 8.47T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 - 0.726T + 89T^{2} \) |
| 97 | \( 1 - 11.8T + 97T^{2} \) |
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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
−8.875467211403504728917778751407, −8.225955085733880397163286002205, −7.32476285340707755301994066983, −6.32907481919067956776282492032, −5.87680631211959321384431624315, −4.92783370320219690686548802241, −3.85226739441753691322872889388, −2.88162775284480920000795486535, −1.90661556510589595599555199899, −0.65448817269444542584941897440,
0.65448817269444542584941897440, 1.90661556510589595599555199899, 2.88162775284480920000795486535, 3.85226739441753691322872889388, 4.92783370320219690686548802241, 5.87680631211959321384431624315, 6.32907481919067956776282492032, 7.32476285340707755301994066983, 8.225955085733880397163286002205, 8.875467211403504728917778751407
