| L(s) = 1 | − 3-s + 4·7-s + 9-s − 4·13-s + 6·17-s − 19-s − 4·21-s + 6·23-s − 5·25-s − 27-s + 6·29-s − 2·31-s − 4·37-s + 4·39-s + 6·41-s + 4·43-s − 6·47-s + 9·49-s − 6·51-s + 6·53-s + 57-s + 12·59-s + 14·61-s + 4·63-s − 8·67-s − 6·69-s + 14·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.51·7-s + 1/3·9-s − 1.10·13-s + 1.45·17-s − 0.229·19-s − 0.872·21-s + 1.25·23-s − 25-s − 0.192·27-s + 1.11·29-s − 0.359·31-s − 0.657·37-s + 0.640·39-s + 0.937·41-s + 0.609·43-s − 0.875·47-s + 9/7·49-s − 0.840·51-s + 0.824·53-s + 0.132·57-s + 1.56·59-s + 1.79·61-s + 0.503·63-s − 0.977·67-s − 0.722·69-s + 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.542973644\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.542973644\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−10.22249875858310914550450943665, −9.359231481850041434452307896895, −8.211124874872632719716882271254, −7.63111837112808582138120144993, −6.75095657741783063390476290272, −5.38776924769841264609355327167, −5.07485672472001477729499979724, −3.94709989510610133111766841641, −2.40790679424770113535591807077, −1.09773072794770812951577890272,
1.09773072794770812951577890272, 2.40790679424770113535591807077, 3.94709989510610133111766841641, 5.07485672472001477729499979724, 5.38776924769841264609355327167, 6.75095657741783063390476290272, 7.63111837112808582138120144993, 8.211124874872632719716882271254, 9.359231481850041434452307896895, 10.22249875858310914550450943665
