| L(s) = 1 | + 3.45·3-s + 5-s + 4.18·7-s + 8.96·9-s − 1.91·11-s + 5.31·13-s + 3.45·15-s − 6.96·17-s − 19-s + 14.4·21-s − 2.55·23-s + 25-s + 20.6·27-s − 2.90·29-s + 4.26·31-s − 6.63·33-s + 4.18·35-s − 6.07·37-s + 18.3·39-s − 6.91·41-s − 3.64·43-s + 8.96·45-s + 6.29·47-s + 10.5·49-s − 24.0·51-s + 12.7·53-s − 1.91·55-s + ⋯ |
| L(s) = 1 | + 1.99·3-s + 0.447·5-s + 1.58·7-s + 2.98·9-s − 0.578·11-s + 1.47·13-s + 0.893·15-s − 1.68·17-s − 0.229·19-s + 3.15·21-s − 0.533·23-s + 0.200·25-s + 3.96·27-s − 0.539·29-s + 0.766·31-s − 1.15·33-s + 0.707·35-s − 0.999·37-s + 2.94·39-s − 1.08·41-s − 0.556·43-s + 1.33·45-s + 0.917·47-s + 1.50·49-s − 3.37·51-s + 1.74·53-s − 0.258·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.035470325\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.035470325\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 3.45T + 3T^{2} \) |
| 7 | \( 1 - 4.18T + 7T^{2} \) |
| 11 | \( 1 + 1.91T + 11T^{2} \) |
| 13 | \( 1 - 5.31T + 13T^{2} \) |
| 17 | \( 1 + 6.96T + 17T^{2} \) |
| 23 | \( 1 + 2.55T + 23T^{2} \) |
| 29 | \( 1 + 2.90T + 29T^{2} \) |
| 31 | \( 1 - 4.26T + 31T^{2} \) |
| 37 | \( 1 + 6.07T + 37T^{2} \) |
| 41 | \( 1 + 6.91T + 41T^{2} \) |
| 43 | \( 1 + 3.64T + 43T^{2} \) |
| 47 | \( 1 - 6.29T + 47T^{2} \) |
| 53 | \( 1 - 12.7T + 53T^{2} \) |
| 59 | \( 1 + 5.24T + 59T^{2} \) |
| 61 | \( 1 - 3.37T + 61T^{2} \) |
| 67 | \( 1 + 8.20T + 67T^{2} \) |
| 71 | \( 1 + 16.3T + 71T^{2} \) |
| 73 | \( 1 + 3.30T + 73T^{2} \) |
| 79 | \( 1 + 1.02T + 79T^{2} \) |
| 83 | \( 1 + 4.34T + 83T^{2} \) |
| 89 | \( 1 + 1.45T + 89T^{2} \) |
| 97 | \( 1 + 8.51T + 97T^{2} \) |
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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
−8.347848135039228143030570951336, −7.55090784545056495589273255606, −6.96985266593514642896197770108, −6.00621552774000934652638685075, −4.87652979032261644436767061768, −4.31224204510337163619006014664, −3.61482390338805504454729189795, −2.59802015547729433316790380890, −1.94239544962094570968932147477, −1.39500721012441549276771296797,
1.39500721012441549276771296797, 1.94239544962094570968932147477, 2.59802015547729433316790380890, 3.61482390338805504454729189795, 4.31224204510337163619006014664, 4.87652979032261644436767061768, 6.00621552774000934652638685075, 6.96985266593514642896197770108, 7.55090784545056495589273255606, 8.347848135039228143030570951336
