| L(s) = 1 | − 3.64·2-s + 5.26·4-s − 14.5·5-s + 6.09·7-s + 9.94·8-s + 53.1·10-s + 58.4·11-s − 28.5·13-s − 22.1·14-s − 78.3·16-s + 43.4·17-s − 97.1·19-s − 76.8·20-s − 212.·22-s + 47.0·23-s + 87.9·25-s + 103.·26-s + 32.0·28-s − 35.8·29-s + 114.·31-s + 205.·32-s − 158.·34-s − 88.8·35-s + 185.·37-s + 354.·38-s − 145.·40-s − 179.·41-s + ⋯ |
| L(s) = 1 | − 1.28·2-s + 0.658·4-s − 1.30·5-s + 0.328·7-s + 0.439·8-s + 1.68·10-s + 1.60·11-s − 0.608·13-s − 0.423·14-s − 1.22·16-s + 0.619·17-s − 1.17·19-s − 0.859·20-s − 2.06·22-s + 0.426·23-s + 0.703·25-s + 0.784·26-s + 0.216·28-s − 0.229·29-s + 0.660·31-s + 1.13·32-s − 0.798·34-s − 0.429·35-s + 0.823·37-s + 1.51·38-s − 0.573·40-s − 0.683·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 239 | \( 1 + 239T \) |
| good | 2 | \( 1 + 3.64T + 8T^{2} \) |
| 5 | \( 1 + 14.5T + 125T^{2} \) |
| 7 | \( 1 - 6.09T + 343T^{2} \) |
| 11 | \( 1 - 58.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 28.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 43.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 97.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 47.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 35.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 114.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 185.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 179.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 303.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 385.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 6.48T + 1.48e5T^{2} \) |
| 59 | \( 1 - 673.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 778.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 102.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 634.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 541.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 235.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 503.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 713.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 479.T + 9.12e5T^{2} \) |
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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
−8.201552576722226082180709372728, −7.957637769598819410982036550400, −6.97258043371449070492211338604, −6.46321838697286790412387409713, −4.89690306021666614815444738855, −4.24139420166245003406068106857, −3.38863994309886883869482236544, −1.92242688360497139451964969740, −0.959877509373789659868092104680, 0,
0.959877509373789659868092104680, 1.92242688360497139451964969740, 3.38863994309886883869482236544, 4.24139420166245003406068106857, 4.89690306021666614815444738855, 6.46321838697286790412387409713, 6.97258043371449070492211338604, 7.957637769598819410982036550400, 8.201552576722226082180709372728
