| L(s) = 1 | + 2.25·2-s + 3.10·4-s − 1.54·5-s + 7-s + 2.49·8-s − 3.48·10-s − 2.53·13-s + 2.25·14-s − 0.568·16-s − 6.59·17-s + 7.56·19-s − 4.79·20-s + 5.61·23-s − 2.61·25-s − 5.73·26-s + 3.10·28-s − 5.45·29-s − 5.32·31-s − 6.27·32-s − 14.9·34-s − 1.54·35-s − 6.18·37-s + 17.0·38-s − 3.85·40-s + 9.78·41-s − 9.61·43-s + 12.6·46-s + ⋯ |
| L(s) = 1 | + 1.59·2-s + 1.55·4-s − 0.690·5-s + 0.377·7-s + 0.882·8-s − 1.10·10-s − 0.703·13-s + 0.603·14-s − 0.142·16-s − 1.59·17-s + 1.73·19-s − 1.07·20-s + 1.17·23-s − 0.523·25-s − 1.12·26-s + 0.586·28-s − 1.01·29-s − 0.956·31-s − 1.10·32-s − 2.55·34-s − 0.260·35-s − 1.01·37-s + 2.77·38-s − 0.609·40-s + 1.52·41-s − 1.46·43-s + 1.87·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.25T + 2T^{2} \) |
| 5 | \( 1 + 1.54T + 5T^{2} \) |
| 13 | \( 1 + 2.53T + 13T^{2} \) |
| 17 | \( 1 + 6.59T + 17T^{2} \) |
| 19 | \( 1 - 7.56T + 19T^{2} \) |
| 23 | \( 1 - 5.61T + 23T^{2} \) |
| 29 | \( 1 + 5.45T + 29T^{2} \) |
| 31 | \( 1 + 5.32T + 31T^{2} \) |
| 37 | \( 1 + 6.18T + 37T^{2} \) |
| 41 | \( 1 - 9.78T + 41T^{2} \) |
| 43 | \( 1 + 9.61T + 43T^{2} \) |
| 47 | \( 1 - 3.84T + 47T^{2} \) |
| 53 | \( 1 - 0.531T + 53T^{2} \) |
| 59 | \( 1 + 1.76T + 59T^{2} \) |
| 61 | \( 1 - 4.74T + 61T^{2} \) |
| 67 | \( 1 + 9.60T + 67T^{2} \) |
| 71 | \( 1 + 9.94T + 71T^{2} \) |
| 73 | \( 1 + 9.71T + 73T^{2} \) |
| 79 | \( 1 + 2.07T + 79T^{2} \) |
| 83 | \( 1 - 0.0442T + 83T^{2} \) |
| 89 | \( 1 - 6.38T + 89T^{2} \) |
| 97 | \( 1 + 13.2T + 97T^{2} \) |
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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
−7.17912190239776742522175955936, −6.95234101374855391649122592923, −5.81260822112606971101936938153, −5.29783327730974624850783164861, −4.65382913254750552678270089085, −4.04951032041906553900041103856, −3.31887559123735234745742753916, −2.60289801593650593467566360046, −1.64272534035368895527096009408, 0,
1.64272534035368895527096009408, 2.60289801593650593467566360046, 3.31887559123735234745742753916, 4.04951032041906553900041103856, 4.65382913254750552678270089085, 5.29783327730974624850783164861, 5.81260822112606971101936938153, 6.95234101374855391649122592923, 7.17912190239776742522175955936
