| L(s) = 1 | + 3·3-s + 6·5-s − 7·7-s + 9·9-s − 36·11-s + 62·13-s + 18·15-s + 114·17-s + 76·19-s − 21·21-s + 24·23-s − 89·25-s + 27·27-s + 54·29-s + 112·31-s − 108·33-s − 42·35-s − 178·37-s + 186·39-s + 378·41-s + 172·43-s + 54·45-s + 192·47-s + 49·49-s + 342·51-s − 402·53-s − 216·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.536·5-s − 0.377·7-s + 1/3·9-s − 0.986·11-s + 1.32·13-s + 0.309·15-s + 1.62·17-s + 0.917·19-s − 0.218·21-s + 0.217·23-s − 0.711·25-s + 0.192·27-s + 0.345·29-s + 0.648·31-s − 0.569·33-s − 0.202·35-s − 0.790·37-s + 0.763·39-s + 1.43·41-s + 0.609·43-s + 0.178·45-s + 0.595·47-s + 1/7·49-s + 0.939·51-s − 1.04·53-s − 0.529·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.584556628\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.584556628\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 7 | \( 1 + p T \) |
| good | 5 | \( 1 - 6 T + p^{3} T^{2} \) |
| 11 | \( 1 + 36 T + p^{3} T^{2} \) |
| 13 | \( 1 - 62 T + p^{3} T^{2} \) |
| 17 | \( 1 - 114 T + p^{3} T^{2} \) |
| 19 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 23 | \( 1 - 24 T + p^{3} T^{2} \) |
| 29 | \( 1 - 54 T + p^{3} T^{2} \) |
| 31 | \( 1 - 112 T + p^{3} T^{2} \) |
| 37 | \( 1 + 178 T + p^{3} T^{2} \) |
| 41 | \( 1 - 378 T + p^{3} T^{2} \) |
| 43 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 47 | \( 1 - 192 T + p^{3} T^{2} \) |
| 53 | \( 1 + 402 T + p^{3} T^{2} \) |
| 59 | \( 1 + 396 T + p^{3} T^{2} \) |
| 61 | \( 1 - 254 T + p^{3} T^{2} \) |
| 67 | \( 1 - 1012 T + p^{3} T^{2} \) |
| 71 | \( 1 + 840 T + p^{3} T^{2} \) |
| 73 | \( 1 - 890 T + p^{3} T^{2} \) |
| 79 | \( 1 + 80 T + p^{3} T^{2} \) |
| 83 | \( 1 - 108 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1638 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1010 T + p^{3} T^{2} \) |
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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
−10.95003132218027914624160392671, −10.06199692065523417038801183107, −9.376676641742990537415439140887, −8.246181489396146654819448721091, −7.50284072013868361508503537521, −6.14755796492914405524142349963, −5.31223783818746135762871179897, −3.71488541907174818961242162607, −2.73054156029581730321426397168, −1.16238814542127758889712844928,
1.16238814542127758889712844928, 2.73054156029581730321426397168, 3.71488541907174818961242162607, 5.31223783818746135762871179897, 6.14755796492914405524142349963, 7.50284072013868361508503537521, 8.246181489396146654819448721091, 9.376676641742990537415439140887, 10.06199692065523417038801183107, 10.95003132218027914624160392671
