| L(s) = 1 | + 2.13·3-s + 0.257·5-s − 0.578·7-s + 1.54·9-s + 1.44·11-s + 1.23·13-s + 0.548·15-s + 3.22·17-s + 2.16·19-s − 1.23·21-s + 3.59·23-s − 4.93·25-s − 3.10·27-s + 1.34·29-s − 2.91·31-s + 3.07·33-s − 0.148·35-s + 7.43·37-s + 2.62·39-s + 9.61·41-s + 4.00·43-s + 0.397·45-s + 7.55·47-s − 6.66·49-s + 6.88·51-s − 5.08·53-s + 0.371·55-s + ⋯ |
| L(s) = 1 | + 1.23·3-s + 0.115·5-s − 0.218·7-s + 0.514·9-s + 0.435·11-s + 0.342·13-s + 0.141·15-s + 0.783·17-s + 0.496·19-s − 0.268·21-s + 0.748·23-s − 0.986·25-s − 0.597·27-s + 0.248·29-s − 0.523·31-s + 0.535·33-s − 0.0251·35-s + 1.22·37-s + 0.420·39-s + 1.50·41-s + 0.610·43-s + 0.0592·45-s + 1.10·47-s − 0.952·49-s + 0.964·51-s − 0.698·53-s + 0.0500·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.247840773\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.247840773\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 - 2.13T + 3T^{2} \) |
| 5 | \( 1 - 0.257T + 5T^{2} \) |
| 7 | \( 1 + 0.578T + 7T^{2} \) |
| 11 | \( 1 - 1.44T + 11T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 17 | \( 1 - 3.22T + 17T^{2} \) |
| 19 | \( 1 - 2.16T + 19T^{2} \) |
| 23 | \( 1 - 3.59T + 23T^{2} \) |
| 29 | \( 1 - 1.34T + 29T^{2} \) |
| 31 | \( 1 + 2.91T + 31T^{2} \) |
| 37 | \( 1 - 7.43T + 37T^{2} \) |
| 41 | \( 1 - 9.61T + 41T^{2} \) |
| 43 | \( 1 - 4.00T + 43T^{2} \) |
| 47 | \( 1 - 7.55T + 47T^{2} \) |
| 53 | \( 1 + 5.08T + 53T^{2} \) |
| 59 | \( 1 - 4.79T + 59T^{2} \) |
| 61 | \( 1 + 0.119T + 61T^{2} \) |
| 67 | \( 1 - 8.76T + 67T^{2} \) |
| 71 | \( 1 + 14.2T + 71T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 + 14.7T + 79T^{2} \) |
| 83 | \( 1 + 9.70T + 83T^{2} \) |
| 89 | \( 1 - 15.5T + 89T^{2} \) |
| 97 | \( 1 - 17.4T + 97T^{2} \) |
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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
−8.466962391563079260070640493661, −7.73063472883338515216100332070, −7.29235937157878090851391258419, −6.17868459331718566039952510336, −5.60691913304579515122358351869, −4.43852535700573488647234130354, −3.65172019864347158540738578495, −3.00163687324987865973890406689, −2.15047859355783359128818561100, −1.01873155319082119995949406872,
1.01873155319082119995949406872, 2.15047859355783359128818561100, 3.00163687324987865973890406689, 3.65172019864347158540738578495, 4.43852535700573488647234130354, 5.60691913304579515122358351869, 6.17868459331718566039952510336, 7.29235937157878090851391258419, 7.73063472883338515216100332070, 8.466962391563079260070640493661
