| L(s) = 1 | + 2·5-s − 6·11-s − 4·13-s + 17-s + 2·19-s + 6·23-s − 25-s − 8·29-s + 4·31-s + 2·37-s + 10·41-s − 4·43-s − 12·47-s − 2·53-s − 12·55-s + 12·59-s + 14·61-s − 8·65-s − 4·67-s − 14·71-s − 6·73-s − 8·79-s − 4·83-s + 2·85-s − 2·89-s + 4·95-s + 14·97-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.80·11-s − 1.10·13-s + 0.242·17-s + 0.458·19-s + 1.25·23-s − 1/5·25-s − 1.48·29-s + 0.718·31-s + 0.328·37-s + 1.56·41-s − 0.609·43-s − 1.75·47-s − 0.274·53-s − 1.61·55-s + 1.56·59-s + 1.79·61-s − 0.992·65-s − 0.488·67-s − 1.66·71-s − 0.702·73-s − 0.900·79-s − 0.439·83-s + 0.216·85-s − 0.211·89-s + 0.410·95-s + 1.42·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.65245498719631, −13.17366483322607, −12.96048219006437, −12.67196443522077, −11.66280569082390, −11.52497772139257, −10.81597500290696, −10.30306057519553, −9.821135637950346, −9.677024401890959, −8.981541789993331, −8.352625969868488, −7.839370393446717, −7.312846099866024, −7.040074043444009, −6.161371309458205, −5.623674643305611, −5.308247902641039, −4.832946252944214, −4.231499663241081, −3.229338449740438, −2.859246609010784, −2.279766330965124, −1.761818864794364, −0.8062903177179745, 0,
0.8062903177179745, 1.761818864794364, 2.279766330965124, 2.859246609010784, 3.229338449740438, 4.231499663241081, 4.832946252944214, 5.308247902641039, 5.623674643305611, 6.161371309458205, 7.040074043444009, 7.312846099866024, 7.839370393446717, 8.352625969868488, 8.981541789993331, 9.677024401890959, 9.821135637950346, 10.30306057519553, 10.81597500290696, 11.52497772139257, 11.66280569082390, 12.67196443522077, 12.96048219006437, 13.17366483322607, 13.65245498719631
