| L(s) = 1 | + 3-s + 3.19·5-s + 1.45·7-s + 9-s − 2.08·11-s − 6.56·13-s + 3.19·15-s − 6.27·17-s − 5.02·19-s + 1.45·21-s − 0.747·23-s + 5.20·25-s + 27-s − 2.93·29-s − 2.63·31-s − 2.08·33-s + 4.65·35-s − 4.91·37-s − 6.56·39-s − 7.59·41-s + 7.77·43-s + 3.19·45-s + 9.47·47-s − 4.88·49-s − 6.27·51-s − 5.61·53-s − 6.65·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.42·5-s + 0.550·7-s + 0.333·9-s − 0.627·11-s − 1.82·13-s + 0.824·15-s − 1.52·17-s − 1.15·19-s + 0.317·21-s − 0.155·23-s + 1.04·25-s + 0.192·27-s − 0.544·29-s − 0.473·31-s − 0.362·33-s + 0.785·35-s − 0.808·37-s − 1.05·39-s − 1.18·41-s + 1.18·43-s + 0.476·45-s + 1.38·47-s − 0.697·49-s − 0.878·51-s − 0.771·53-s − 0.896·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 127 | \( 1 - T \) |
| good | 5 | \( 1 - 3.19T + 5T^{2} \) |
| 7 | \( 1 - 1.45T + 7T^{2} \) |
| 11 | \( 1 + 2.08T + 11T^{2} \) |
| 13 | \( 1 + 6.56T + 13T^{2} \) |
| 17 | \( 1 + 6.27T + 17T^{2} \) |
| 19 | \( 1 + 5.02T + 19T^{2} \) |
| 23 | \( 1 + 0.747T + 23T^{2} \) |
| 29 | \( 1 + 2.93T + 29T^{2} \) |
| 31 | \( 1 + 2.63T + 31T^{2} \) |
| 37 | \( 1 + 4.91T + 37T^{2} \) |
| 41 | \( 1 + 7.59T + 41T^{2} \) |
| 43 | \( 1 - 7.77T + 43T^{2} \) |
| 47 | \( 1 - 9.47T + 47T^{2} \) |
| 53 | \( 1 + 5.61T + 53T^{2} \) |
| 59 | \( 1 + 1.83T + 59T^{2} \) |
| 61 | \( 1 - 5.89T + 61T^{2} \) |
| 67 | \( 1 + 8.33T + 67T^{2} \) |
| 71 | \( 1 + 9.95T + 71T^{2} \) |
| 73 | \( 1 + 2.08T + 73T^{2} \) |
| 79 | \( 1 - 8.73T + 79T^{2} \) |
| 83 | \( 1 - 3.49T + 83T^{2} \) |
| 89 | \( 1 + 8.16T + 89T^{2} \) |
| 97 | \( 1 - 0.808T + 97T^{2} \) |
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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
−7.66529161639958159101537395089, −7.06298493000534586720131643695, −6.33382212004751600603694338499, −5.48107194719722323655226417719, −4.85236916955471445592200431983, −4.22137970252866347729911464610, −2.86849158292185937996037589266, −2.14844534091935779181908049324, −1.86649066559093895730568207807, 0,
1.86649066559093895730568207807, 2.14844534091935779181908049324, 2.86849158292185937996037589266, 4.22137970252866347729911464610, 4.85236916955471445592200431983, 5.48107194719722323655226417719, 6.33382212004751600603694338499, 7.06298493000534586720131643695, 7.66529161639958159101537395089
