| L(s) = 1 | − 2-s + 1.93·3-s + 4-s − 1.45·5-s − 1.93·6-s + 4.05·7-s − 8-s + 0.755·9-s + 1.45·10-s − 0.983·11-s + 1.93·12-s − 4.24·13-s − 4.05·14-s − 2.82·15-s + 16-s + 5.70·17-s − 0.755·18-s − 2.48·19-s − 1.45·20-s + 7.86·21-s + 0.983·22-s − 1.99·23-s − 1.93·24-s − 2.86·25-s + 4.24·26-s − 4.34·27-s + 4.05·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.11·3-s + 0.5·4-s − 0.652·5-s − 0.791·6-s + 1.53·7-s − 0.353·8-s + 0.251·9-s + 0.461·10-s − 0.296·11-s + 0.559·12-s − 1.17·13-s − 1.08·14-s − 0.730·15-s + 0.250·16-s + 1.38·17-s − 0.178·18-s − 0.569·19-s − 0.326·20-s + 1.71·21-s + 0.209·22-s − 0.416·23-s − 0.395·24-s − 0.573·25-s + 0.833·26-s − 0.836·27-s + 0.767·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 2011 | \( 1 + T \) |
| good | 3 | \( 1 - 1.93T + 3T^{2} \) |
| 5 | \( 1 + 1.45T + 5T^{2} \) |
| 7 | \( 1 - 4.05T + 7T^{2} \) |
| 11 | \( 1 + 0.983T + 11T^{2} \) |
| 13 | \( 1 + 4.24T + 13T^{2} \) |
| 17 | \( 1 - 5.70T + 17T^{2} \) |
| 19 | \( 1 + 2.48T + 19T^{2} \) |
| 23 | \( 1 + 1.99T + 23T^{2} \) |
| 29 | \( 1 + 9.80T + 29T^{2} \) |
| 31 | \( 1 + 7.00T + 31T^{2} \) |
| 37 | \( 1 + 8.98T + 37T^{2} \) |
| 41 | \( 1 - 4.21T + 41T^{2} \) |
| 43 | \( 1 + 2.46T + 43T^{2} \) |
| 47 | \( 1 - 4.12T + 47T^{2} \) |
| 53 | \( 1 + 10.0T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 + 0.603T + 61T^{2} \) |
| 67 | \( 1 + 4.56T + 67T^{2} \) |
| 71 | \( 1 + 3.76T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 - 5.55T + 79T^{2} \) |
| 83 | \( 1 - 11.9T + 83T^{2} \) |
| 89 | \( 1 - 8.04T + 89T^{2} \) |
| 97 | \( 1 - 11.1T + 97T^{2} \) |
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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
−7.935932234689006019022274911190, −7.67158438139163294348661762658, −7.25980600775924284987908963506, −5.73473798007853544712654872253, −5.11900706935143921584603044246, −4.03447498680422191593578948340, −3.33168517502027389101042022630, −2.21196141715711587920177833415, −1.69385318520258521122030921021, 0,
1.69385318520258521122030921021, 2.21196141715711587920177833415, 3.33168517502027389101042022630, 4.03447498680422191593578948340, 5.11900706935143921584603044246, 5.73473798007853544712654872253, 7.25980600775924284987908963506, 7.67158438139163294348661762658, 7.935932234689006019022274911190
