| L(s) = 1 | + 3-s − 3.34·5-s − 2.51·7-s + 9-s + 11-s + 13-s − 3.34·15-s + 7.59·17-s − 1.48·19-s − 2.51·21-s − 1.55·23-s + 6.21·25-s + 27-s + 1.86·29-s − 1.61·31-s + 33-s + 8.43·35-s + 6.43·37-s + 39-s − 10.9·41-s − 5.59·43-s − 3.34·45-s + 0.697·47-s − 0.663·49-s + 7.59·51-s − 9.73·53-s − 3.34·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.49·5-s − 0.951·7-s + 0.333·9-s + 0.301·11-s + 0.277·13-s − 0.864·15-s + 1.84·17-s − 0.340·19-s − 0.549·21-s − 0.323·23-s + 1.24·25-s + 0.192·27-s + 0.346·29-s − 0.290·31-s + 0.174·33-s + 1.42·35-s + 1.05·37-s + 0.160·39-s − 1.70·41-s − 0.853·43-s − 0.499·45-s + 0.101·47-s − 0.0947·49-s + 1.06·51-s − 1.33·53-s − 0.451·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 3.34T + 5T^{2} \) |
| 7 | \( 1 + 2.51T + 7T^{2} \) |
| 17 | \( 1 - 7.59T + 17T^{2} \) |
| 19 | \( 1 + 1.48T + 19T^{2} \) |
| 23 | \( 1 + 1.55T + 23T^{2} \) |
| 29 | \( 1 - 1.86T + 29T^{2} \) |
| 31 | \( 1 + 1.61T + 31T^{2} \) |
| 37 | \( 1 - 6.43T + 37T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 + 5.59T + 43T^{2} \) |
| 47 | \( 1 - 0.697T + 47T^{2} \) |
| 53 | \( 1 + 9.73T + 53T^{2} \) |
| 59 | \( 1 + 4.76T + 59T^{2} \) |
| 61 | \( 1 + 1.66T + 61T^{2} \) |
| 67 | \( 1 - 4.81T + 67T^{2} \) |
| 71 | \( 1 + 0.965T + 71T^{2} \) |
| 73 | \( 1 + 3.21T + 73T^{2} \) |
| 79 | \( 1 + 8.83T + 79T^{2} \) |
| 83 | \( 1 + 0.767T + 83T^{2} \) |
| 89 | \( 1 - 3.41T + 89T^{2} \) |
| 97 | \( 1 + 6.96T + 97T^{2} \) |
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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
−8.096257963477105190068555778035, −7.69288206810673277677117436300, −6.85601564456220479672150650743, −6.14756595657845276762041135960, −5.03556776514998939357063445750, −4.05345281391183321264713316014, −3.46396870575626944621706727400, −2.94193545304530127941615945433, −1.35455937392424887287462384832, 0,
1.35455937392424887287462384832, 2.94193545304530127941615945433, 3.46396870575626944621706727400, 4.05345281391183321264713316014, 5.03556776514998939357063445750, 6.14756595657845276762041135960, 6.85601564456220479672150650743, 7.69288206810673277677117436300, 8.096257963477105190068555778035
