| L(s) = 1 | − 2-s + 4-s + 3.41·5-s − 7-s − 8-s − 3.41·10-s − 4.24·11-s + 0.828·13-s + 14-s + 16-s + 0.828·17-s − 7.07·19-s + 3.41·20-s + 4.24·22-s − 23-s + 6.65·25-s − 0.828·26-s − 28-s − 2·29-s + 0.828·31-s − 32-s − 0.828·34-s − 3.41·35-s − 10.2·37-s + 7.07·38-s − 3.41·40-s + 2·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.52·5-s − 0.377·7-s − 0.353·8-s − 1.07·10-s − 1.27·11-s + 0.229·13-s + 0.267·14-s + 0.250·16-s + 0.200·17-s − 1.62·19-s + 0.763·20-s + 0.904·22-s − 0.208·23-s + 1.33·25-s − 0.162·26-s − 0.188·28-s − 0.371·29-s + 0.148·31-s − 0.176·32-s − 0.142·34-s − 0.577·35-s − 1.68·37-s + 1.14·38-s − 0.539·40-s + 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 3.41T + 5T^{2} \) |
| 11 | \( 1 + 4.24T + 11T^{2} \) |
| 13 | \( 1 - 0.828T + 13T^{2} \) |
| 17 | \( 1 - 0.828T + 17T^{2} \) |
| 19 | \( 1 + 7.07T + 19T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 0.828T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 6.58T + 43T^{2} \) |
| 47 | \( 1 - 3.17T + 47T^{2} \) |
| 53 | \( 1 - 1.75T + 53T^{2} \) |
| 59 | \( 1 + 2.82T + 59T^{2} \) |
| 61 | \( 1 + 5.75T + 61T^{2} \) |
| 67 | \( 1 + 3.75T + 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 + 12T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 - 6.58T + 83T^{2} \) |
| 89 | \( 1 - 3.65T + 89T^{2} \) |
| 97 | \( 1 + 3.17T + 97T^{2} \) |
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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
−8.665420122511873194221137429715, −7.69427887546498788651324037730, −6.85705098928990260666994201663, −6.07458389481775354609824617623, −5.61064726870416570315531641604, −4.62361348362088067946248612563, −3.22685378033512367368277082804, −2.33760420069190936204504625492, −1.64723041859288584136467029380, 0,
1.64723041859288584136467029380, 2.33760420069190936204504625492, 3.22685378033512367368277082804, 4.62361348362088067946248612563, 5.61064726870416570315531641604, 6.07458389481775354609824617623, 6.85705098928990260666994201663, 7.69427887546498788651324037730, 8.665420122511873194221137429715
