L(s) = 1 | + 2-s + 2.40·3-s + 4-s + 1.76·5-s + 2.40·6-s − 7-s + 8-s + 2.79·9-s + 1.76·10-s + 1.18·11-s + 2.40·12-s + 0.361·13-s − 14-s + 4.24·15-s + 16-s + 3.54·17-s + 2.79·18-s + 1.76·20-s − 2.40·21-s + 1.18·22-s − 2.87·23-s + 2.40·24-s − 1.89·25-s + 0.361·26-s − 0.489·27-s − 28-s + 5.10·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.39·3-s + 0.5·4-s + 0.787·5-s + 0.982·6-s − 0.377·7-s + 0.353·8-s + 0.932·9-s + 0.557·10-s + 0.357·11-s + 0.695·12-s + 0.100·13-s − 0.267·14-s + 1.09·15-s + 0.250·16-s + 0.859·17-s + 0.659·18-s + 0.393·20-s − 0.525·21-s + 0.252·22-s − 0.599·23-s + 0.491·24-s − 0.379·25-s + 0.0709·26-s − 0.0942·27-s − 0.188·28-s + 0.947·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.180431306\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.180431306\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 - 2.40T + 3T^{2} \) |
| 5 | \( 1 - 1.76T + 5T^{2} \) |
| 11 | \( 1 - 1.18T + 11T^{2} \) |
| 13 | \( 1 - 0.361T + 13T^{2} \) |
| 17 | \( 1 - 3.54T + 17T^{2} \) |
| 23 | \( 1 + 2.87T + 23T^{2} \) |
| 29 | \( 1 - 5.10T + 29T^{2} \) |
| 31 | \( 1 + 1.38T + 31T^{2} \) |
| 37 | \( 1 + 3.35T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 - 3.47T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 - 12.0T + 53T^{2} \) |
| 59 | \( 1 + 7.72T + 59T^{2} \) |
| 61 | \( 1 + 1.99T + 61T^{2} \) |
| 67 | \( 1 - 8.30T + 67T^{2} \) |
| 71 | \( 1 + 0.226T + 71T^{2} \) |
| 73 | \( 1 + 14.0T + 73T^{2} \) |
| 79 | \( 1 - 5.28T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 + 1.87T + 89T^{2} \) |
| 97 | \( 1 + 3.99T + 97T^{2} \) |
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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
−8.142850612621540918653527364474, −7.58327614551130216997300396148, −6.78964246321469075775159659258, −5.94087193581048102976044554088, −5.45759383983685373476744865716, −4.20612792526149849465646620532, −3.73269111248161254977692109573, −2.78484396637408625749808664576, −2.30013285996831195968224095864, −1.25753981066687820218593731211,
1.25753981066687820218593731211, 2.30013285996831195968224095864, 2.78484396637408625749808664576, 3.73269111248161254977692109573, 4.20612792526149849465646620532, 5.45759383983685373476744865716, 5.94087193581048102976044554088, 6.78964246321469075775159659258, 7.58327614551130216997300396148, 8.142850612621540918653527364474