| L(s) = 1 | + 3-s − 2·4-s + 2·7-s + 9-s + 11-s − 2·12-s − 13-s + 4·16-s + 3·17-s + 19-s + 2·21-s + 6·23-s − 5·25-s + 27-s − 4·28-s + 8·31-s + 33-s − 2·36-s + 2·37-s − 39-s + 6·41-s + 8·43-s − 2·44-s − 6·47-s + 4·48-s − 3·49-s + 3·51-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s + 0.755·7-s + 1/3·9-s + 0.301·11-s − 0.577·12-s − 0.277·13-s + 16-s + 0.727·17-s + 0.229·19-s + 0.436·21-s + 1.25·23-s − 25-s + 0.192·27-s − 0.755·28-s + 1.43·31-s + 0.174·33-s − 1/3·36-s + 0.328·37-s − 0.160·39-s + 0.937·41-s + 1.21·43-s − 0.301·44-s − 0.875·47-s + 0.577·48-s − 3/7·49-s + 0.420·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 627 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 627 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.631559008\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.631559008\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 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
−10.38316548933611880460422425874, −9.595676059271080215098446016156, −8.885855868195721527212596861827, −8.047058109228960310569118011279, −7.38987844970363499694447516693, −5.92615105889325146373690957476, −4.88303727227954019406472250536, −4.10497206474885287550986629629, −2.89395743647375258219896560652, −1.21678682746529038884045056230,
1.21678682746529038884045056230, 2.89395743647375258219896560652, 4.10497206474885287550986629629, 4.88303727227954019406472250536, 5.92615105889325146373690957476, 7.38987844970363499694447516693, 8.047058109228960310569118011279, 8.885855868195721527212596861827, 9.595676059271080215098446016156, 10.38316548933611880460422425874
