| L(s) = 1 | + 3-s − 5-s − 0.688·7-s + 9-s + 3.80·11-s + 0.688·13-s − 15-s − 6.70·17-s − 4·19-s − 0.688·21-s − 4.14·23-s + 25-s + 27-s − 0.969·29-s − 31-s + 3.80·33-s + 0.688·35-s − 5.73·37-s + 0.688·39-s − 7.05·41-s − 6·43-s − 45-s + 10.3·47-s − 6.52·49-s − 6.70·51-s + 12.3·53-s − 3.80·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.260·7-s + 0.333·9-s + 1.14·11-s + 0.191·13-s − 0.258·15-s − 1.62·17-s − 0.917·19-s − 0.150·21-s − 0.864·23-s + 0.200·25-s + 0.192·27-s − 0.180·29-s − 0.179·31-s + 0.662·33-s + 0.116·35-s − 0.943·37-s + 0.110·39-s − 1.10·41-s − 0.914·43-s − 0.149·45-s + 1.51·47-s − 0.932·49-s − 0.939·51-s + 1.70·53-s − 0.513·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 + 0.688T + 7T^{2} \) |
| 11 | \( 1 - 3.80T + 11T^{2} \) |
| 13 | \( 1 - 0.688T + 13T^{2} \) |
| 17 | \( 1 + 6.70T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 4.14T + 23T^{2} \) |
| 29 | \( 1 + 0.969T + 29T^{2} \) |
| 37 | \( 1 + 5.73T + 37T^{2} \) |
| 41 | \( 1 + 7.05T + 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 - 12.3T + 53T^{2} \) |
| 59 | \( 1 + 4.77T + 59T^{2} \) |
| 61 | \( 1 + 6.29T + 61T^{2} \) |
| 67 | \( 1 + 0.453T + 67T^{2} \) |
| 71 | \( 1 + 3.45T + 71T^{2} \) |
| 73 | \( 1 - 13.3T + 73T^{2} \) |
| 79 | \( 1 + 8.08T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 + 14.0T + 89T^{2} \) |
| 97 | \( 1 + 0.949T + 97T^{2} \) |
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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
−8.413863727817522931801965410586, −7.36097149053848718245271331555, −6.69313864168683522447517182013, −6.18342265027214143319285998954, −4.93475692060602402190712101662, −4.05647716367706533062004713914, −3.66850047532899530806151205911, −2.46360337790271577488418113296, −1.60260897076899393711042393453, 0,
1.60260897076899393711042393453, 2.46360337790271577488418113296, 3.66850047532899530806151205911, 4.05647716367706533062004713914, 4.93475692060602402190712101662, 6.18342265027214143319285998954, 6.69313864168683522447517182013, 7.36097149053848718245271331555, 8.413863727817522931801965410586
