| L(s) = 1 | − 2-s + 4-s + 4·5-s − 7-s − 8-s − 4·10-s − 11-s − 13-s + 14-s + 16-s − 6·19-s + 4·20-s + 22-s + 7·23-s + 11·25-s + 26-s − 28-s + 4·29-s + 7·31-s − 32-s − 4·35-s + 9·37-s + 6·38-s − 4·40-s + 3·41-s + 4·43-s − 44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.78·5-s − 0.377·7-s − 0.353·8-s − 1.26·10-s − 0.301·11-s − 0.277·13-s + 0.267·14-s + 1/4·16-s − 1.37·19-s + 0.894·20-s + 0.213·22-s + 1.45·23-s + 11/5·25-s + 0.196·26-s − 0.188·28-s + 0.742·29-s + 1.25·31-s − 0.176·32-s − 0.676·35-s + 1.47·37-s + 0.973·38-s − 0.632·40-s + 0.468·41-s + 0.609·43-s − 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.650233464\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.650233464\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−9.382683634490519654764199003796, −8.842134311662064724454282147488, −7.943345217051475117643242740083, −6.72942118020662831970006205526, −6.37840271109207786796552495603, −5.49386334296658824475642157597, −4.55256625494296783763579353487, −2.85733179130965228288861874054, −2.28863422577859859560688974167, −1.03297262712619069700335338528,
1.03297262712619069700335338528, 2.28863422577859859560688974167, 2.85733179130965228288861874054, 4.55256625494296783763579353487, 5.49386334296658824475642157597, 6.37840271109207786796552495603, 6.72942118020662831970006205526, 7.943345217051475117643242740083, 8.842134311662064724454282147488, 9.382683634490519654764199003796
