| L(s) = 1 | + 3·2-s + 6.72·3-s + 4-s + 20.1·6-s − 26.6·7-s − 21·8-s + 18.1·9-s − 39.6·11-s + 6.72·12-s + 23.1·13-s − 79.8·14-s − 71·16-s + 2.95·17-s + 54.4·18-s + 32.3·19-s − 178.·21-s − 118.·22-s + 23·23-s − 141.·24-s + 69.4·26-s − 59.4·27-s − 26.6·28-s − 162.·29-s − 241.·31-s − 45·32-s − 266.·33-s + 8.87·34-s + ⋯ |
| L(s) = 1 | + 1.06·2-s + 1.29·3-s + 0.125·4-s + 1.37·6-s − 1.43·7-s − 0.928·8-s + 0.672·9-s − 1.08·11-s + 0.161·12-s + 0.494·13-s − 1.52·14-s − 1.10·16-s + 0.0422·17-s + 0.713·18-s + 0.390·19-s − 1.85·21-s − 1.15·22-s + 0.208·23-s − 1.20·24-s + 0.524·26-s − 0.423·27-s − 0.179·28-s − 1.04·29-s − 1.39·31-s − 0.248·32-s − 1.40·33-s + 0.0447·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 2 | \( 1 - 3T + 8T^{2} \) |
| 3 | \( 1 - 6.72T + 27T^{2} \) |
| 7 | \( 1 + 26.6T + 343T^{2} \) |
| 11 | \( 1 + 39.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 23.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 2.95T + 4.91e3T^{2} \) |
| 19 | \( 1 - 32.3T + 6.85e3T^{2} \) |
| 29 | \( 1 + 162.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 241.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 180.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 353.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 365.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 195.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 461.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 290.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 301.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 366.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 8.14T + 3.57e5T^{2} \) |
| 73 | \( 1 + 360.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.24e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.48e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 829.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 390.T + 9.12e5T^{2} \) |
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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
−9.596790750579772027340815673745, −9.127690540215575037758700125931, −8.162973405205794854082720029086, −7.12907747086111907569338545971, −6.03774187067438412374647181610, −5.15307007544042008112551042493, −3.68941453978593320115912225746, −3.32074386865370223429479011662, −2.34339234615357796550160343163, 0,
2.34339234615357796550160343163, 3.32074386865370223429479011662, 3.68941453978593320115912225746, 5.15307007544042008112551042493, 6.03774187067438412374647181610, 7.12907747086111907569338545971, 8.162973405205794854082720029086, 9.127690540215575037758700125931, 9.596790750579772027340815673745
