L(s) = 1 | − 2-s + 3-s + 4-s − 0.387·5-s − 6-s + 5.18·7-s − 8-s + 9-s + 0.387·10-s − 0.897·11-s + 12-s − 13-s − 5.18·14-s − 0.387·15-s + 16-s + 3.72·17-s − 18-s − 4.53·19-s − 0.387·20-s + 5.18·21-s + 0.897·22-s + 9.05·23-s − 24-s − 4.84·25-s + 26-s + 27-s + 5.18·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.173·5-s − 0.408·6-s + 1.96·7-s − 0.353·8-s + 0.333·9-s + 0.122·10-s − 0.270·11-s + 0.288·12-s − 0.277·13-s − 1.38·14-s − 0.100·15-s + 0.250·16-s + 0.902·17-s − 0.235·18-s − 1.03·19-s − 0.0866·20-s + 1.13·21-s + 0.191·22-s + 1.88·23-s − 0.204·24-s − 0.969·25-s + 0.196·26-s + 0.192·27-s + 0.980·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.406636462\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.406636462\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
good | 5 | \( 1 + 0.387T + 5T^{2} \) |
| 7 | \( 1 - 5.18T + 7T^{2} \) |
| 11 | \( 1 + 0.897T + 11T^{2} \) |
| 17 | \( 1 - 3.72T + 17T^{2} \) |
| 19 | \( 1 + 4.53T + 19T^{2} \) |
| 23 | \( 1 - 9.05T + 23T^{2} \) |
| 29 | \( 1 + 2.78T + 29T^{2} \) |
| 31 | \( 1 - 7.73T + 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 - 3.40T + 41T^{2} \) |
| 43 | \( 1 + 11.2T + 43T^{2} \) |
| 47 | \( 1 - 3.73T + 47T^{2} \) |
| 53 | \( 1 - 1.47T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 - 13.5T + 61T^{2} \) |
| 67 | \( 1 + 5.48T + 67T^{2} \) |
| 71 | \( 1 - 12.8T + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 + 16.6T + 79T^{2} \) |
| 83 | \( 1 + 6.67T + 83T^{2} \) |
| 89 | \( 1 + 2.89T + 89T^{2} \) |
| 97 | \( 1 - 5.04T + 97T^{2} \) |
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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
−8.130472107954907240959093697803, −7.26134564641199433592457532659, −6.88941062590640675798201324006, −5.59211063871874316265339072906, −5.07045267969370941214412912063, −4.31259571809091396320959795156, −3.40102569599119330066763214625, −2.38646805822258268791762822383, −1.76068475987454794961348839554, −0.871184321219939816477837531347,
0.871184321219939816477837531347, 1.76068475987454794961348839554, 2.38646805822258268791762822383, 3.40102569599119330066763214625, 4.31259571809091396320959795156, 5.07045267969370941214412912063, 5.59211063871874316265339072906, 6.88941062590640675798201324006, 7.26134564641199433592457532659, 8.130472107954907240959093697803