L(s) = 1 | + 2-s + 4-s + 2.08·5-s − 4.01·7-s + 8-s + 2.08·10-s + 0.212·11-s + 5.19·13-s − 4.01·14-s + 16-s + 3.78·17-s − 1.27·19-s + 2.08·20-s + 0.212·22-s + 0.829·23-s − 0.640·25-s + 5.19·26-s − 4.01·28-s + 9.21·29-s + 3.53·31-s + 32-s + 3.78·34-s − 8.37·35-s + 3.55·37-s − 1.27·38-s + 2.08·40-s + 2.73·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.933·5-s − 1.51·7-s + 0.353·8-s + 0.660·10-s + 0.0640·11-s + 1.44·13-s − 1.07·14-s + 0.250·16-s + 0.918·17-s − 0.292·19-s + 0.466·20-s + 0.0452·22-s + 0.172·23-s − 0.128·25-s + 1.01·26-s − 0.758·28-s + 1.71·29-s + 0.635·31-s + 0.176·32-s + 0.649·34-s − 1.41·35-s + 0.584·37-s − 0.206·38-s + 0.330·40-s + 0.427·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.919798375\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.919798375\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 5 | \( 1 - 2.08T + 5T^{2} \) |
| 7 | \( 1 + 4.01T + 7T^{2} \) |
| 11 | \( 1 - 0.212T + 11T^{2} \) |
| 13 | \( 1 - 5.19T + 13T^{2} \) |
| 17 | \( 1 - 3.78T + 17T^{2} \) |
| 19 | \( 1 + 1.27T + 19T^{2} \) |
| 23 | \( 1 - 0.829T + 23T^{2} \) |
| 29 | \( 1 - 9.21T + 29T^{2} \) |
| 31 | \( 1 - 3.53T + 31T^{2} \) |
| 37 | \( 1 - 3.55T + 37T^{2} \) |
| 41 | \( 1 - 2.73T + 41T^{2} \) |
| 43 | \( 1 - 6.89T + 43T^{2} \) |
| 47 | \( 1 + 1.27T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 + 8.55T + 59T^{2} \) |
| 61 | \( 1 - 2.83T + 61T^{2} \) |
| 67 | \( 1 - 9.76T + 67T^{2} \) |
| 71 | \( 1 + 11.7T + 71T^{2} \) |
| 73 | \( 1 - 0.750T + 73T^{2} \) |
| 79 | \( 1 + 2.36T + 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 - 5.39T + 89T^{2} \) |
| 97 | \( 1 + 12.5T + 97T^{2} \) |
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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
−9.681323210473444963365884814597, −8.842439038297019987054871833043, −7.83431070202804399223317891245, −6.59522117469654784081390590544, −6.22801853423355438937807711276, −5.62818016914827362843431646778, −4.37179189783946259388407730577, −3.37680789233657773873145152989, −2.67168166473271919511339058394, −1.20275109850105654258291118253,
1.20275109850105654258291118253, 2.67168166473271919511339058394, 3.37680789233657773873145152989, 4.37179189783946259388407730577, 5.62818016914827362843431646778, 6.22801853423355438937807711276, 6.59522117469654784081390590544, 7.83431070202804399223317891245, 8.842439038297019987054871833043, 9.681323210473444963365884814597