| L(s) = 1 | + 2-s + 4-s + 2·5-s + 8-s + 2·10-s + 13-s + 16-s − 17-s + 2·20-s − 25-s + 26-s + 2·29-s + 8·31-s + 32-s − 34-s + 10·37-s + 2·40-s + 6·41-s − 4·43-s + 8·47-s − 7·49-s − 50-s + 52-s + 2·53-s + 2·58-s + 4·59-s − 10·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.353·8-s + 0.632·10-s + 0.277·13-s + 1/4·16-s − 0.242·17-s + 0.447·20-s − 1/5·25-s + 0.196·26-s + 0.371·29-s + 1.43·31-s + 0.176·32-s − 0.171·34-s + 1.64·37-s + 0.316·40-s + 0.937·41-s − 0.609·43-s + 1.16·47-s − 49-s − 0.141·50-s + 0.138·52-s + 0.274·53-s + 0.262·58-s + 0.520·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3978 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3978 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.836383042\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.836383042\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−8.349075713869114964617851030605, −7.69848340965912357010192540613, −6.67658910432642137189037903452, −6.19526021335418308400003649059, −5.53265123890702070276610187771, −4.68728429723521121006940415248, −3.97045121261786692214956305782, −2.87292729806150489106541111120, −2.19892207420390815799712979776, −1.06689533391547720689125103096,
1.06689533391547720689125103096, 2.19892207420390815799712979776, 2.87292729806150489106541111120, 3.97045121261786692214956305782, 4.68728429723521121006940415248, 5.53265123890702070276610187771, 6.19526021335418308400003649059, 6.67658910432642137189037903452, 7.69848340965912357010192540613, 8.349075713869114964617851030605
