| L(s) = 1 | + 2·5-s − 4.24·7-s − 3·9-s + 4.24·11-s + 13-s + 4·17-s + 4.24·19-s − 25-s + 2·29-s − 4.24·31-s − 8.48·35-s + 6·37-s + 2·41-s − 8.48·43-s − 6·45-s − 12.7·47-s + 10.9·49-s + 8·53-s + 8.48·55-s − 4.24·59-s + 12.7·63-s + 2·65-s + 4.24·67-s − 4.24·71-s + 6·73-s − 17.9·77-s + 8.48·79-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.60·7-s − 9-s + 1.27·11-s + 0.277·13-s + 0.970·17-s + 0.973·19-s − 0.200·25-s + 0.371·29-s − 0.762·31-s − 1.43·35-s + 0.986·37-s + 0.312·41-s − 1.29·43-s − 0.894·45-s − 1.85·47-s + 1.57·49-s + 1.09·53-s + 1.14·55-s − 0.552·59-s + 1.60·63-s + 0.248·65-s + 0.518·67-s − 0.503·71-s + 0.702·73-s − 2.05·77-s + 0.954·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.825196358\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.825196358\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 7 | \( 1 + 4.24T + 7T^{2} \) |
| 11 | \( 1 - 4.24T + 11T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 - 4.24T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 4.24T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 8.48T + 43T^{2} \) |
| 47 | \( 1 + 12.7T + 47T^{2} \) |
| 53 | \( 1 - 8T + 53T^{2} \) |
| 59 | \( 1 + 4.24T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 4.24T + 67T^{2} \) |
| 71 | \( 1 + 4.24T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 8.48T + 79T^{2} \) |
| 83 | \( 1 - 12.7T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 18T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.880065425571858706942799890627, −7.906473759734572903941838466636, −6.92523747964179791876431093472, −6.16410290635155308074101184602, −5.94246221186174764485190134229, −4.96840338041671652676169804616, −3.46556431099378018558028922610, −3.33491292055218823431806026057, −2.07705917948157762578129462099, −0.806189440881641009930242873914,
0.806189440881641009930242873914, 2.07705917948157762578129462099, 3.33491292055218823431806026057, 3.46556431099378018558028922610, 4.96840338041671652676169804616, 5.94246221186174764485190134229, 6.16410290635155308074101184602, 6.92523747964179791876431093472, 7.906473759734572903941838466636, 8.880065425571858706942799890627
