| L(s) = 1 | + 2.17·2-s − 0.995·3-s + 2.73·4-s − 2.16·6-s − 3.57·7-s + 1.58·8-s − 2.00·9-s − 2.14·11-s − 2.71·12-s − 1.64·13-s − 7.77·14-s − 2.00·16-s − 0.660·17-s − 4.37·18-s + 4.16·19-s + 3.55·21-s − 4.66·22-s + 0.236·23-s − 1.58·24-s − 3.57·26-s + 4.98·27-s − 9.75·28-s − 6.17·29-s − 0.0561·31-s − 7.54·32-s + 2.13·33-s − 1.43·34-s + ⋯ |
| L(s) = 1 | + 1.53·2-s − 0.574·3-s + 1.36·4-s − 0.883·6-s − 1.35·7-s + 0.561·8-s − 0.669·9-s − 0.646·11-s − 0.784·12-s − 0.456·13-s − 2.07·14-s − 0.501·16-s − 0.160·17-s − 1.03·18-s + 0.955·19-s + 0.776·21-s − 0.993·22-s + 0.0492·23-s − 0.322·24-s − 0.701·26-s + 0.959·27-s − 1.84·28-s − 1.14·29-s − 0.0100·31-s − 1.33·32-s + 0.371·33-s − 0.246·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.178202187\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.178202187\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 + T \) |
| good | 2 | \( 1 - 2.17T + 2T^{2} \) |
| 3 | \( 1 + 0.995T + 3T^{2} \) |
| 7 | \( 1 + 3.57T + 7T^{2} \) |
| 11 | \( 1 + 2.14T + 11T^{2} \) |
| 13 | \( 1 + 1.64T + 13T^{2} \) |
| 17 | \( 1 + 0.660T + 17T^{2} \) |
| 19 | \( 1 - 4.16T + 19T^{2} \) |
| 23 | \( 1 - 0.236T + 23T^{2} \) |
| 29 | \( 1 + 6.17T + 29T^{2} \) |
| 31 | \( 1 + 0.0561T + 31T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 - 9.65T + 41T^{2} \) |
| 43 | \( 1 - 11.8T + 43T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 + 0.824T + 53T^{2} \) |
| 59 | \( 1 - 0.481T + 59T^{2} \) |
| 61 | \( 1 - 15.0T + 61T^{2} \) |
| 67 | \( 1 - 1.76T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 - 7.30T + 83T^{2} \) |
| 89 | \( 1 - 8.65T + 89T^{2} \) |
| 97 | \( 1 - 2.57T + 97T^{2} \) |
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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
−7.63930986298625076348114866562, −7.17112756799917940069063282421, −6.23804290463653696859887793101, −5.78095996237781697456515565113, −5.41435280913649930328735483852, −4.43203722152216432621920952154, −3.77972373268012389844512104026, −2.77018420504885643759981209083, −2.58111948073556780738318893735, −0.59073045465297120631767020341,
0.59073045465297120631767020341, 2.58111948073556780738318893735, 2.77018420504885643759981209083, 3.77972373268012389844512104026, 4.43203722152216432621920952154, 5.41435280913649930328735483852, 5.78095996237781697456515565113, 6.23804290463653696859887793101, 7.17112756799917940069063282421, 7.63930986298625076348114866562
