| L(s) = 1 | − 2-s + 4-s + 1.09·5-s + 1.60·7-s − 8-s − 1.09·10-s + 0.449·11-s + 2.24·13-s − 1.60·14-s + 16-s + 7.95·17-s − 3.17·19-s + 1.09·20-s − 0.449·22-s + 6.84·23-s − 3.80·25-s − 2.24·26-s + 1.60·28-s + 2.86·29-s − 5.51·31-s − 32-s − 7.95·34-s + 1.75·35-s + 6.70·37-s + 3.17·38-s − 1.09·40-s − 0.500·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.488·5-s + 0.606·7-s − 0.353·8-s − 0.345·10-s + 0.135·11-s + 0.622·13-s − 0.429·14-s + 0.250·16-s + 1.92·17-s − 0.727·19-s + 0.244·20-s − 0.0958·22-s + 1.42·23-s − 0.761·25-s − 0.440·26-s + 0.303·28-s + 0.532·29-s − 0.990·31-s − 0.176·32-s − 1.36·34-s + 0.296·35-s + 1.10·37-s + 0.514·38-s − 0.172·40-s − 0.0782·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.874468100\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.874468100\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 223 | \( 1 - T \) |
| good | 5 | \( 1 - 1.09T + 5T^{2} \) |
| 7 | \( 1 - 1.60T + 7T^{2} \) |
| 11 | \( 1 - 0.449T + 11T^{2} \) |
| 13 | \( 1 - 2.24T + 13T^{2} \) |
| 17 | \( 1 - 7.95T + 17T^{2} \) |
| 19 | \( 1 + 3.17T + 19T^{2} \) |
| 23 | \( 1 - 6.84T + 23T^{2} \) |
| 29 | \( 1 - 2.86T + 29T^{2} \) |
| 31 | \( 1 + 5.51T + 31T^{2} \) |
| 37 | \( 1 - 6.70T + 37T^{2} \) |
| 41 | \( 1 + 0.500T + 41T^{2} \) |
| 43 | \( 1 - 4.03T + 43T^{2} \) |
| 47 | \( 1 - 4.16T + 47T^{2} \) |
| 53 | \( 1 - 0.688T + 53T^{2} \) |
| 59 | \( 1 - 0.561T + 59T^{2} \) |
| 61 | \( 1 - 3.56T + 61T^{2} \) |
| 67 | \( 1 + 4.84T + 67T^{2} \) |
| 71 | \( 1 + 1.34T + 71T^{2} \) |
| 73 | \( 1 + 4.50T + 73T^{2} \) |
| 79 | \( 1 + 7.27T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 + 10.1T + 89T^{2} \) |
| 97 | \( 1 - 0.701T + 97T^{2} \) |
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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
−8.562619767900816722809969191456, −7.70403173955064899626452643921, −7.24877617113314641031778459961, −6.14561721039347630573737078560, −5.70585333268208604084025787936, −4.77915686073488015979524284157, −3.70739215385475464625968942862, −2.80680311259483166773625579936, −1.71796351349531185244051562129, −0.944148765458794331083254455413,
0.944148765458794331083254455413, 1.71796351349531185244051562129, 2.80680311259483166773625579936, 3.70739215385475464625968942862, 4.77915686073488015979524284157, 5.70585333268208604084025787936, 6.14561721039347630573737078560, 7.24877617113314641031778459961, 7.70403173955064899626452643921, 8.562619767900816722809969191456
