| L(s) = 1 | − 2.41·2-s − 3-s + 3.82·4-s + 2.41·6-s − 2·7-s − 4.41·8-s + 9-s + 2.82·11-s − 3.82·12-s + 4.82·13-s + 4.82·14-s + 2.99·16-s − 3.65·17-s − 2.41·18-s − 1.24·19-s + 2·21-s − 6.82·22-s − 4.41·23-s + 4.41·24-s − 11.6·26-s − 27-s − 7.65·28-s − 4.82·29-s + 6·31-s + 1.58·32-s − 2.82·33-s + 8.82·34-s + ⋯ |
| L(s) = 1 | − 1.70·2-s − 0.577·3-s + 1.91·4-s + 0.985·6-s − 0.755·7-s − 1.56·8-s + 0.333·9-s + 0.852·11-s − 1.10·12-s + 1.33·13-s + 1.29·14-s + 0.749·16-s − 0.886·17-s − 0.569·18-s − 0.285·19-s + 0.436·21-s − 1.45·22-s − 0.920·23-s + 0.901·24-s − 2.28·26-s − 0.192·27-s − 1.44·28-s − 0.896·29-s + 1.07·31-s + 0.280·32-s − 0.492·33-s + 1.51·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 - 4.82T + 13T^{2} \) |
| 17 | \( 1 + 3.65T + 17T^{2} \) |
| 19 | \( 1 + 1.24T + 19T^{2} \) |
| 23 | \( 1 + 4.41T + 23T^{2} \) |
| 29 | \( 1 + 4.82T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 41 | \( 1 - 0.656T + 41T^{2} \) |
| 43 | \( 1 - 1.24T + 43T^{2} \) |
| 47 | \( 1 + 8.82T + 47T^{2} \) |
| 53 | \( 1 - 2.65T + 53T^{2} \) |
| 59 | \( 1 - 7.58T + 59T^{2} \) |
| 61 | \( 1 + 12T + 61T^{2} \) |
| 67 | \( 1 - 7.65T + 67T^{2} \) |
| 71 | \( 1 + 7.31T + 71T^{2} \) |
| 73 | \( 1 - 11.4T + 73T^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 - 1.65T + 83T^{2} \) |
| 89 | \( 1 + 7.17T + 89T^{2} \) |
| 97 | \( 1 + 8.48T + 97T^{2} \) |
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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
−8.509119163775860223081886789828, −7.933386544779355930789985285318, −6.77316594138352322046681957239, −6.53660570146667884239437736086, −5.82160151096562051334979389482, −4.37522343039994769610771653971, −3.45817831846979806314241159398, −2.11960394745079395442384258373, −1.16611427232106977742660714343, 0,
1.16611427232106977742660714343, 2.11960394745079395442384258373, 3.45817831846979806314241159398, 4.37522343039994769610771653971, 5.82160151096562051334979389482, 6.53660570146667884239437736086, 6.77316594138352322046681957239, 7.933386544779355930789985285318, 8.509119163775860223081886789828
