| L(s) = 1 | − 0.467·3-s − 5-s − 0.199·7-s − 2.78·9-s − 2.26·11-s − 4.04·13-s + 0.467·15-s + 3.44·17-s + 3.75·19-s + 0.0930·21-s − 2.08·23-s + 25-s + 2.70·27-s − 0.834·29-s − 6.59·31-s + 1.05·33-s + 0.199·35-s + 0.220·37-s + 1.89·39-s + 0.751·41-s − 4.57·43-s + 2.78·45-s − 3.31·47-s − 6.96·49-s − 1.60·51-s − 2.32·53-s + 2.26·55-s + ⋯ |
| L(s) = 1 | − 0.269·3-s − 0.447·5-s − 0.0752·7-s − 0.927·9-s − 0.683·11-s − 1.12·13-s + 0.120·15-s + 0.835·17-s + 0.861·19-s + 0.0203·21-s − 0.434·23-s + 0.200·25-s + 0.519·27-s − 0.154·29-s − 1.18·31-s + 0.184·33-s + 0.0336·35-s + 0.0363·37-s + 0.302·39-s + 0.117·41-s − 0.697·43-s + 0.414·45-s − 0.483·47-s − 0.994·49-s − 0.225·51-s − 0.319·53-s + 0.305·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7853824432\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7853824432\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 151 | \( 1 - T \) |
| good | 3 | \( 1 + 0.467T + 3T^{2} \) |
| 7 | \( 1 + 0.199T + 7T^{2} \) |
| 11 | \( 1 + 2.26T + 11T^{2} \) |
| 13 | \( 1 + 4.04T + 13T^{2} \) |
| 17 | \( 1 - 3.44T + 17T^{2} \) |
| 19 | \( 1 - 3.75T + 19T^{2} \) |
| 23 | \( 1 + 2.08T + 23T^{2} \) |
| 29 | \( 1 + 0.834T + 29T^{2} \) |
| 31 | \( 1 + 6.59T + 31T^{2} \) |
| 37 | \( 1 - 0.220T + 37T^{2} \) |
| 41 | \( 1 - 0.751T + 41T^{2} \) |
| 43 | \( 1 + 4.57T + 43T^{2} \) |
| 47 | \( 1 + 3.31T + 47T^{2} \) |
| 53 | \( 1 + 2.32T + 53T^{2} \) |
| 59 | \( 1 + 7.96T + 59T^{2} \) |
| 61 | \( 1 - 7.63T + 61T^{2} \) |
| 67 | \( 1 + 4.27T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 + 10.0T + 73T^{2} \) |
| 79 | \( 1 + 2.08T + 79T^{2} \) |
| 83 | \( 1 - 6.41T + 83T^{2} \) |
| 89 | \( 1 + 15.5T + 89T^{2} \) |
| 97 | \( 1 - 18.2T + 97T^{2} \) |
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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
−7.87953288635618821131736573117, −7.55542454953176185186112910595, −6.70877941990172024467503943801, −5.76524936887719800716390716009, −5.27000766956517130864198191117, −4.61758816588265152809738576579, −3.43936497258060745552184407896, −2.95104343339652191722849008120, −1.89119088535350945858078415221, −0.45063224972327728119954091462,
0.45063224972327728119954091462, 1.89119088535350945858078415221, 2.95104343339652191722849008120, 3.43936497258060745552184407896, 4.61758816588265152809738576579, 5.27000766956517130864198191117, 5.76524936887719800716390716009, 6.70877941990172024467503943801, 7.55542454953176185186112910595, 7.87953288635618821131736573117
