| L(s) = 1 | + 0.470·2-s − 3-s − 1.77·4-s + 5-s − 0.470·6-s − 7-s − 1.77·8-s + 9-s + 0.470·10-s − 11-s + 1.77·12-s − 0.249·13-s − 0.470·14-s − 15-s + 2.71·16-s + 1.22·17-s + 0.470·18-s − 1.83·19-s − 1.77·20-s + 21-s − 0.470·22-s + 3.71·23-s + 1.77·24-s + 25-s − 0.117·26-s − 27-s + 1.77·28-s + ⋯ |
| L(s) = 1 | + 0.332·2-s − 0.577·3-s − 0.889·4-s + 0.447·5-s − 0.192·6-s − 0.377·7-s − 0.628·8-s + 0.333·9-s + 0.148·10-s − 0.301·11-s + 0.513·12-s − 0.0690·13-s − 0.125·14-s − 0.258·15-s + 0.679·16-s + 0.296·17-s + 0.110·18-s − 0.421·19-s − 0.397·20-s + 0.218·21-s − 0.100·22-s + 0.775·23-s + 0.363·24-s + 0.200·25-s − 0.0229·26-s − 0.192·27-s + 0.336·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.181582322\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.181582322\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 0.470T + 2T^{2} \) |
| 13 | \( 1 + 0.249T + 13T^{2} \) |
| 17 | \( 1 - 1.22T + 17T^{2} \) |
| 19 | \( 1 + 1.83T + 19T^{2} \) |
| 23 | \( 1 - 3.71T + 23T^{2} \) |
| 29 | \( 1 + 3.96T + 29T^{2} \) |
| 31 | \( 1 - 7.80T + 31T^{2} \) |
| 37 | \( 1 - 4.24T + 37T^{2} \) |
| 41 | \( 1 - 3.19T + 41T^{2} \) |
| 43 | \( 1 - 3.02T + 43T^{2} \) |
| 47 | \( 1 + 1.19T + 47T^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 59 | \( 1 - 3.02T + 59T^{2} \) |
| 61 | \( 1 - 9.15T + 61T^{2} \) |
| 67 | \( 1 - 7.67T + 67T^{2} \) |
| 71 | \( 1 + 4.13T + 71T^{2} \) |
| 73 | \( 1 + 0.941T + 73T^{2} \) |
| 79 | \( 1 + 8.36T + 79T^{2} \) |
| 83 | \( 1 - 1.10T + 83T^{2} \) |
| 89 | \( 1 - 5.26T + 89T^{2} \) |
| 97 | \( 1 - 11.1T + 97T^{2} \) |
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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
−9.852565288641253504480747612739, −9.091501767603384801557876446371, −8.260524602104659586100969422671, −7.18288237160315258489033032213, −6.18931276150015006518314951753, −5.51938416753831974711773950632, −4.71657754385482775448862296121, −3.79571584770304697644406656620, −2.58964167009184430126048890280, −0.811708100176887385856908714038,
0.811708100176887385856908714038, 2.58964167009184430126048890280, 3.79571584770304697644406656620, 4.71657754385482775448862296121, 5.51938416753831974711773950632, 6.18931276150015006518314951753, 7.18288237160315258489033032213, 8.260524602104659586100969422671, 9.091501767603384801557876446371, 9.852565288641253504480747612739
