| L(s) = 1 | + 3-s − 0.473·5-s + 0.663·7-s + 9-s + 3.46·11-s − 2.43·13-s − 0.473·15-s − 5.99·17-s − 2.40·19-s + 0.663·21-s − 8.17·23-s − 4.77·25-s + 27-s + 3.87·29-s − 2.70·31-s + 3.46·33-s − 0.314·35-s + 8.48·37-s − 2.43·39-s + 3.71·41-s − 10.5·43-s − 0.473·45-s − 10.9·47-s − 6.55·49-s − 5.99·51-s + 0.259·53-s − 1.64·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.211·5-s + 0.250·7-s + 0.333·9-s + 1.04·11-s − 0.673·13-s − 0.122·15-s − 1.45·17-s − 0.550·19-s + 0.144·21-s − 1.70·23-s − 0.955·25-s + 0.192·27-s + 0.719·29-s − 0.486·31-s + 0.603·33-s − 0.0531·35-s + 1.39·37-s − 0.389·39-s + 0.580·41-s − 1.60·43-s − 0.0706·45-s − 1.59·47-s − 0.937·49-s − 0.839·51-s + 0.0357·53-s − 0.221·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 + 0.473T + 5T^{2} \) |
| 7 | \( 1 - 0.663T + 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 + 2.43T + 13T^{2} \) |
| 17 | \( 1 + 5.99T + 17T^{2} \) |
| 19 | \( 1 + 2.40T + 19T^{2} \) |
| 23 | \( 1 + 8.17T + 23T^{2} \) |
| 29 | \( 1 - 3.87T + 29T^{2} \) |
| 31 | \( 1 + 2.70T + 31T^{2} \) |
| 37 | \( 1 - 8.48T + 37T^{2} \) |
| 41 | \( 1 - 3.71T + 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 - 0.259T + 53T^{2} \) |
| 59 | \( 1 + 1.16T + 59T^{2} \) |
| 61 | \( 1 + 1.55T + 61T^{2} \) |
| 67 | \( 1 + 1.52T + 67T^{2} \) |
| 71 | \( 1 + 4.46T + 71T^{2} \) |
| 73 | \( 1 + 7.44T + 73T^{2} \) |
| 79 | \( 1 - 8.68T + 79T^{2} \) |
| 83 | \( 1 - 15.7T + 83T^{2} \) |
| 89 | \( 1 + 5.88T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−8.158558821950084021051116666908, −7.47735732787606476935940622165, −6.54418493841571413758750020210, −6.12901989534384933651128324384, −4.77179535365337229207107596371, −4.27721790590279422552743450436, −3.50233815210036930219859530438, −2.34152839061225936286118665823, −1.67927871121022525576417319446, 0,
1.67927871121022525576417319446, 2.34152839061225936286118665823, 3.50233815210036930219859530438, 4.27721790590279422552743450436, 4.77179535365337229207107596371, 6.12901989534384933651128324384, 6.54418493841571413758750020210, 7.47735732787606476935940622165, 8.158558821950084021051116666908
