| L(s) = 1 | + 3-s − 5-s + 4.27·7-s + 9-s − 2.27·11-s + 13-s − 15-s − 6.27·17-s − 2·19-s + 4.27·21-s − 0.274·23-s + 25-s + 27-s − 4·31-s − 2.27·33-s − 4.27·35-s + 8.27·37-s + 39-s + 8.27·41-s + 8.54·43-s − 45-s − 4·47-s + 11.2·49-s − 6.27·51-s + 8.27·53-s + 2.27·55-s − 2·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.61·7-s + 0.333·9-s − 0.685·11-s + 0.277·13-s − 0.258·15-s − 1.52·17-s − 0.458·19-s + 0.932·21-s − 0.0573·23-s + 0.200·25-s + 0.192·27-s − 0.718·31-s − 0.396·33-s − 0.722·35-s + 1.36·37-s + 0.160·39-s + 1.29·41-s + 1.30·43-s − 0.149·45-s − 0.583·47-s + 1.61·49-s − 0.878·51-s + 1.13·53-s + 0.306·55-s − 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.626215911\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.626215911\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 4.27T + 7T^{2} \) |
| 11 | \( 1 + 2.27T + 11T^{2} \) |
| 17 | \( 1 + 6.27T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 0.274T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 8.27T + 37T^{2} \) |
| 41 | \( 1 - 8.27T + 41T^{2} \) |
| 43 | \( 1 - 8.54T + 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 - 8.27T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 8.82T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 1.72T + 71T^{2} \) |
| 73 | \( 1 + 12.5T + 73T^{2} \) |
| 79 | \( 1 - 14.2T + 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + 8.27T + 89T^{2} \) |
| 97 | \( 1 - 14.2T + 97T^{2} \) |
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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
−7.915947884505530081412905957107, −7.66621051313389819134354180422, −6.80582981778869299577616377884, −5.86451950337102013706290600931, −4.98740437676365457130244701578, −4.38270658226033688227741020504, −3.81839437954308307575953530378, −2.49234298038442952783859989092, −2.08206636646297597689192477757, −0.827001417960182129841298890052,
0.827001417960182129841298890052, 2.08206636646297597689192477757, 2.49234298038442952783859989092, 3.81839437954308307575953530378, 4.38270658226033688227741020504, 4.98740437676365457130244701578, 5.86451950337102013706290600931, 6.80582981778869299577616377884, 7.66621051313389819134354180422, 7.915947884505530081412905957107
