| L(s) = 1 | + 1.76·3-s + 5-s + 4.62·7-s + 0.103·9-s − 5.52·11-s − 5.49·13-s + 1.76·15-s − 6.62·17-s − 19-s + 8.14·21-s − 4.14·23-s + 25-s − 5.10·27-s + 7.87·29-s − 1.25·31-s − 9.72·33-s + 4.62·35-s + 0.387·37-s − 9.67·39-s + 6.77·41-s − 10.9·43-s + 0.103·45-s − 1.72·47-s + 14.4·49-s − 11.6·51-s − 1.49·53-s − 5.52·55-s + ⋯ |
| L(s) = 1 | + 1.01·3-s + 0.447·5-s + 1.74·7-s + 0.0343·9-s − 1.66·11-s − 1.52·13-s + 0.454·15-s − 1.60·17-s − 0.229·19-s + 1.77·21-s − 0.865·23-s + 0.200·25-s − 0.982·27-s + 1.46·29-s − 0.224·31-s − 1.69·33-s + 0.781·35-s + 0.0637·37-s − 1.54·39-s + 1.05·41-s − 1.67·43-s + 0.0153·45-s − 0.252·47-s + 2.05·49-s − 1.63·51-s − 0.204·53-s − 0.744·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 1.76T + 3T^{2} \) |
| 7 | \( 1 - 4.62T + 7T^{2} \) |
| 11 | \( 1 + 5.52T + 11T^{2} \) |
| 13 | \( 1 + 5.49T + 13T^{2} \) |
| 17 | \( 1 + 6.62T + 17T^{2} \) |
| 23 | \( 1 + 4.14T + 23T^{2} \) |
| 29 | \( 1 - 7.87T + 29T^{2} \) |
| 31 | \( 1 + 1.25T + 31T^{2} \) |
| 37 | \( 1 - 0.387T + 37T^{2} \) |
| 41 | \( 1 - 6.77T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 + 1.72T + 47T^{2} \) |
| 53 | \( 1 + 1.49T + 53T^{2} \) |
| 59 | \( 1 - 0.626T + 59T^{2} \) |
| 61 | \( 1 + 15.0T + 61T^{2} \) |
| 67 | \( 1 - 5.22T + 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 + 4.83T + 73T^{2} \) |
| 79 | \( 1 - 2.98T + 79T^{2} \) |
| 83 | \( 1 + 2.74T + 83T^{2} \) |
| 89 | \( 1 - 4.27T + 89T^{2} \) |
| 97 | \( 1 + 13.7T + 97T^{2} \) |
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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
−7.916346215067665223700755706978, −7.33381280502520934848432098188, −6.33918487124606852072154851258, −5.24433629181362530062002852270, −4.89426276272825863569675698598, −4.21056578965318731338212295740, −2.78272999301739589750607082250, −2.38698190335972048917653610342, −1.78608210830072041461264581885, 0,
1.78608210830072041461264581885, 2.38698190335972048917653610342, 2.78272999301739589750607082250, 4.21056578965318731338212295740, 4.89426276272825863569675698598, 5.24433629181362530062002852270, 6.33918487124606852072154851258, 7.33381280502520934848432098188, 7.916346215067665223700755706978
