| L(s) = 1 | − 3-s − 2.93·5-s − 7-s + 9-s − 5.73·11-s + 2.54·13-s + 2.93·15-s − 2.34·17-s − 19-s + 21-s + 2.32·23-s + 3.60·25-s − 27-s + 6.47·29-s + 9.87·31-s + 5.73·33-s + 2.93·35-s − 3.68·37-s − 2.54·39-s + 10.6·41-s + 4.08·43-s − 2.93·45-s − 4.29·47-s + 49-s + 2.34·51-s + 6.47·53-s + 16.8·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.31·5-s − 0.377·7-s + 0.333·9-s − 1.73·11-s + 0.707·13-s + 0.757·15-s − 0.568·17-s − 0.229·19-s + 0.218·21-s + 0.485·23-s + 0.720·25-s − 0.192·27-s + 1.20·29-s + 1.77·31-s + 0.998·33-s + 0.495·35-s − 0.606·37-s − 0.408·39-s + 1.66·41-s + 0.623·43-s − 0.437·45-s − 0.626·47-s + 0.142·49-s + 0.328·51-s + 0.888·53-s + 2.26·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 2.93T + 5T^{2} \) |
| 11 | \( 1 + 5.73T + 11T^{2} \) |
| 13 | \( 1 - 2.54T + 13T^{2} \) |
| 17 | \( 1 + 2.34T + 17T^{2} \) |
| 23 | \( 1 - 2.32T + 23T^{2} \) |
| 29 | \( 1 - 6.47T + 29T^{2} \) |
| 31 | \( 1 - 9.87T + 31T^{2} \) |
| 37 | \( 1 + 3.68T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 - 4.08T + 43T^{2} \) |
| 47 | \( 1 + 4.29T + 47T^{2} \) |
| 53 | \( 1 - 6.47T + 53T^{2} \) |
| 59 | \( 1 + 7.07T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 2.95T + 67T^{2} \) |
| 71 | \( 1 + 8.98T + 71T^{2} \) |
| 73 | \( 1 - 7.86T + 73T^{2} \) |
| 79 | \( 1 - 2.32T + 79T^{2} \) |
| 83 | \( 1 - 3.29T + 83T^{2} \) |
| 89 | \( 1 + 11.7T + 89T^{2} \) |
| 97 | \( 1 + 1.46T + 97T^{2} \) |
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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
−7.76212504087338323577095833031, −6.98495365015173515437228442807, −6.30882884017573097053978915180, −5.53194108849255752184797004182, −4.64995374377631474963291748151, −4.22262064426861831281566032235, −3.16474926227147986854180538039, −2.52857229689379020122677450956, −0.932033221900979634258681038024, 0,
0.932033221900979634258681038024, 2.52857229689379020122677450956, 3.16474926227147986854180538039, 4.22262064426861831281566032235, 4.64995374377631474963291748151, 5.53194108849255752184797004182, 6.30882884017573097053978915180, 6.98495365015173515437228442807, 7.76212504087338323577095833031
