| L(s) = 1 | − 3-s + 5-s + 1.32·7-s + 9-s − 3.70·11-s + 3.32·13-s − 15-s + 1.61·17-s + 19-s − 1.32·21-s + 7.67·23-s + 25-s − 27-s − 1.70·29-s − 9.67·31-s + 3.70·33-s + 1.32·35-s + 5.96·37-s − 3.32·39-s + 2.29·41-s + 8.73·43-s + 45-s − 11.6·47-s − 5.25·49-s − 1.61·51-s + 10.4·53-s − 3.70·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.499·7-s + 0.333·9-s − 1.11·11-s + 0.921·13-s − 0.258·15-s + 0.391·17-s + 0.229·19-s − 0.288·21-s + 1.59·23-s + 0.200·25-s − 0.192·27-s − 0.317·29-s − 1.73·31-s + 0.645·33-s + 0.223·35-s + 0.980·37-s − 0.531·39-s + 0.358·41-s + 1.33·43-s + 0.149·45-s − 1.70·47-s − 0.750·49-s − 0.225·51-s + 1.43·53-s − 0.499·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.838590180\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.838590180\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 1.32T + 7T^{2} \) |
| 11 | \( 1 + 3.70T + 11T^{2} \) |
| 13 | \( 1 - 3.32T + 13T^{2} \) |
| 17 | \( 1 - 1.61T + 17T^{2} \) |
| 23 | \( 1 - 7.67T + 23T^{2} \) |
| 29 | \( 1 + 1.70T + 29T^{2} \) |
| 31 | \( 1 + 9.67T + 31T^{2} \) |
| 37 | \( 1 - 5.96T + 37T^{2} \) |
| 41 | \( 1 - 2.29T + 41T^{2} \) |
| 43 | \( 1 - 8.73T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 + 12.6T + 59T^{2} \) |
| 61 | \( 1 - 9.02T + 61T^{2} \) |
| 67 | \( 1 - 13.2T + 67T^{2} \) |
| 71 | \( 1 - 8.69T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 - 14.0T + 79T^{2} \) |
| 83 | \( 1 + 5.02T + 83T^{2} \) |
| 89 | \( 1 + 1.70T + 89T^{2} \) |
| 97 | \( 1 + 0.679T + 97T^{2} \) |
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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
−8.237461006035578476351754527824, −7.59110956580293799470602896136, −6.88086063007551704425284640357, −5.98139400138620937695997937572, −5.37755533615438930855914775757, −4.87783835188731916326023182473, −3.80233888997935225496187360522, −2.87160566060930521026877425123, −1.81425456961360794284694198743, −0.804103216101608112698240117450,
0.804103216101608112698240117450, 1.81425456961360794284694198743, 2.87160566060930521026877425123, 3.80233888997935225496187360522, 4.87783835188731916326023182473, 5.37755533615438930855914775757, 5.98139400138620937695997937572, 6.88086063007551704425284640357, 7.59110956580293799470602896136, 8.237461006035578476351754527824
