| L(s) = 1 | − 2-s + 4-s + 4.24·5-s + 0.296·7-s − 8-s − 4.24·10-s + 2.39·11-s + 2.51·13-s − 0.296·14-s + 16-s + 4.50·17-s − 1.13·19-s + 4.24·20-s − 2.39·22-s + 13.0·25-s − 2.51·26-s + 0.296·28-s − 0.957·29-s + 8.24·31-s − 32-s − 4.50·34-s + 1.25·35-s + 8.34·37-s + 1.13·38-s − 4.24·40-s − 4.10·41-s + 3.86·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.89·5-s + 0.111·7-s − 0.353·8-s − 1.34·10-s + 0.722·11-s + 0.697·13-s − 0.0791·14-s + 0.250·16-s + 1.09·17-s − 0.260·19-s + 0.949·20-s − 0.511·22-s + 2.60·25-s − 0.492·26-s + 0.0559·28-s − 0.177·29-s + 1.48·31-s − 0.176·32-s − 0.773·34-s + 0.212·35-s + 1.37·37-s + 0.184·38-s − 0.671·40-s − 0.641·41-s + 0.589·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.909788684\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.909788684\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 - 4.24T + 5T^{2} \) |
| 7 | \( 1 - 0.296T + 7T^{2} \) |
| 11 | \( 1 - 2.39T + 11T^{2} \) |
| 13 | \( 1 - 2.51T + 13T^{2} \) |
| 17 | \( 1 - 4.50T + 17T^{2} \) |
| 19 | \( 1 + 1.13T + 19T^{2} \) |
| 29 | \( 1 + 0.957T + 29T^{2} \) |
| 31 | \( 1 - 8.24T + 31T^{2} \) |
| 37 | \( 1 - 8.34T + 37T^{2} \) |
| 41 | \( 1 + 4.10T + 41T^{2} \) |
| 43 | \( 1 - 3.86T + 43T^{2} \) |
| 47 | \( 1 + 3.35T + 47T^{2} \) |
| 53 | \( 1 + 4.84T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 7.76T + 61T^{2} \) |
| 67 | \( 1 - 14.1T + 67T^{2} \) |
| 71 | \( 1 - 14.7T + 71T^{2} \) |
| 73 | \( 1 + 11.1T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 - 2.12T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 - 0.795T + 97T^{2} \) |
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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
−7.894470782749110760249363854187, −6.70154887580791876796334388744, −6.50072428323362146249833973979, −5.77183298211910365437589760151, −5.23245592440207910186523180744, −4.21126545661793996911905501352, −3.12946983735431254975716661599, −2.43230130900752243090776997910, −1.50486877812528269041285599193, −1.03972492825602015114693622411,
1.03972492825602015114693622411, 1.50486877812528269041285599193, 2.43230130900752243090776997910, 3.12946983735431254975716661599, 4.21126545661793996911905501352, 5.23245592440207910186523180744, 5.77183298211910365437589760151, 6.50072428323362146249833973979, 6.70154887580791876796334388744, 7.894470782749110760249363854187
