| L(s) = 1 | − 2.33·3-s − 5-s − 0.399·7-s + 2.43·9-s + 1.73·11-s + 2.33·15-s + 0.692·17-s − 5.37·19-s + 0.932·21-s − 0.107·23-s + 25-s + 1.30·27-s − 4.90·29-s + 7.86·31-s − 4.03·33-s + 0.399·35-s + 2.26·37-s − 7.73·41-s + 6.01·43-s − 2.43·45-s + 3.46·47-s − 6.84·49-s − 1.61·51-s + 11.7·53-s − 1.73·55-s + 12.5·57-s + 7.27·59-s + ⋯ |
| L(s) = 1 | − 1.34·3-s − 0.447·5-s − 0.151·7-s + 0.813·9-s + 0.522·11-s + 0.602·15-s + 0.167·17-s − 1.23·19-s + 0.203·21-s − 0.0223·23-s + 0.200·25-s + 0.251·27-s − 0.910·29-s + 1.41·31-s − 0.703·33-s + 0.0675·35-s + 0.372·37-s − 1.20·41-s + 0.916·43-s − 0.363·45-s + 0.505·47-s − 0.977·49-s − 0.226·51-s + 1.60·53-s − 0.233·55-s + 1.65·57-s + 0.947·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 2.33T + 3T^{2} \) |
| 7 | \( 1 + 0.399T + 7T^{2} \) |
| 11 | \( 1 - 1.73T + 11T^{2} \) |
| 17 | \( 1 - 0.692T + 17T^{2} \) |
| 19 | \( 1 + 5.37T + 19T^{2} \) |
| 23 | \( 1 + 0.107T + 23T^{2} \) |
| 29 | \( 1 + 4.90T + 29T^{2} \) |
| 31 | \( 1 - 7.86T + 31T^{2} \) |
| 37 | \( 1 - 2.26T + 37T^{2} \) |
| 41 | \( 1 + 7.73T + 41T^{2} \) |
| 43 | \( 1 - 6.01T + 43T^{2} \) |
| 47 | \( 1 - 3.46T + 47T^{2} \) |
| 53 | \( 1 - 11.7T + 53T^{2} \) |
| 59 | \( 1 - 7.27T + 59T^{2} \) |
| 61 | \( 1 - 8.68T + 61T^{2} \) |
| 67 | \( 1 - 1.32T + 67T^{2} \) |
| 71 | \( 1 + 3.87T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 + 13.1T + 79T^{2} \) |
| 83 | \( 1 - 14.0T + 83T^{2} \) |
| 89 | \( 1 - 0.347T + 89T^{2} \) |
| 97 | \( 1 + 8.85T + 97T^{2} \) |
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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
−8.283516698956659777543295476533, −7.27621399035097108172268904851, −6.61504806764341990715151267165, −6.03575085915405888433145720416, −5.26812447162241826503392967940, −4.43223815682522007596046005879, −3.76661470384509717189939341172, −2.48906461017741850908403747312, −1.12585819767334998460404059314, 0,
1.12585819767334998460404059314, 2.48906461017741850908403747312, 3.76661470384509717189939341172, 4.43223815682522007596046005879, 5.26812447162241826503392967940, 6.03575085915405888433145720416, 6.61504806764341990715151267165, 7.27621399035097108172268904851, 8.283516698956659777543295476533
