L(s) = 1 | − 0.401·2-s − 1.83·4-s − 5-s + 2.55·7-s + 1.54·8-s + 0.401·10-s − 6.34·11-s + 13-s − 1.02·14-s + 3.05·16-s + 1.45·17-s − 2.89·19-s + 1.83·20-s + 2.55·22-s + 1.87·23-s + 25-s − 0.401·26-s − 4.70·28-s + 3.57·29-s − 5.67·31-s − 4.31·32-s − 0.583·34-s − 2.55·35-s + 6.18·37-s + 1.16·38-s − 1.54·40-s − 3.33·41-s + ⋯ |
L(s) = 1 | − 0.284·2-s − 0.919·4-s − 0.447·5-s + 0.966·7-s + 0.545·8-s + 0.127·10-s − 1.91·11-s + 0.277·13-s − 0.274·14-s + 0.764·16-s + 0.352·17-s − 0.664·19-s + 0.411·20-s + 0.543·22-s + 0.391·23-s + 0.200·25-s − 0.0787·26-s − 0.888·28-s + 0.664·29-s − 1.01·31-s − 0.762·32-s − 0.100·34-s − 0.432·35-s + 1.01·37-s + 0.188·38-s − 0.243·40-s − 0.520·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9506619690\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9506619690\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + 0.401T + 2T^{2} \) |
| 7 | \( 1 - 2.55T + 7T^{2} \) |
| 11 | \( 1 + 6.34T + 11T^{2} \) |
| 17 | \( 1 - 1.45T + 17T^{2} \) |
| 19 | \( 1 + 2.89T + 19T^{2} \) |
| 23 | \( 1 - 1.87T + 23T^{2} \) |
| 29 | \( 1 - 3.57T + 29T^{2} \) |
| 31 | \( 1 + 5.67T + 31T^{2} \) |
| 37 | \( 1 - 6.18T + 37T^{2} \) |
| 41 | \( 1 + 3.33T + 41T^{2} \) |
| 43 | \( 1 - 9.46T + 43T^{2} \) |
| 47 | \( 1 + 2.77T + 47T^{2} \) |
| 53 | \( 1 - 9.34T + 53T^{2} \) |
| 59 | \( 1 + 11.1T + 59T^{2} \) |
| 61 | \( 1 - 7.83T + 61T^{2} \) |
| 67 | \( 1 - 9.22T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 - 14.8T + 73T^{2} \) |
| 79 | \( 1 - 4.01T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 - 8.56T + 89T^{2} \) |
| 97 | \( 1 + 7.57T + 97T^{2} \) |
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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
−9.161502491011637798753819674656, −8.358156596774056850690569340491, −7.936866772645387606723917487584, −7.31338215429473088129162737133, −5.86459017545886934111228970425, −5.04750866967052751490185098512, −4.53479528483990162612985196699, −3.43812793921502527125826948231, −2.19880451219983957557387237264, −0.69973518528196251035618395044,
0.69973518528196251035618395044, 2.19880451219983957557387237264, 3.43812793921502527125826948231, 4.53479528483990162612985196699, 5.04750866967052751490185098512, 5.86459017545886934111228970425, 7.31338215429473088129162737133, 7.936866772645387606723917487584, 8.358156596774056850690569340491, 9.161502491011637798753819674656