| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s − 4·11-s + 12-s + 13-s + 16-s + 2·17-s − 18-s − 19-s + 4·22-s + 2·23-s − 24-s − 26-s + 27-s + 4·29-s − 32-s − 4·33-s − 2·34-s + 36-s − 3·37-s + 38-s + 39-s − 12·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 1.20·11-s + 0.288·12-s + 0.277·13-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.229·19-s + 0.852·22-s + 0.417·23-s − 0.204·24-s − 0.196·26-s + 0.192·27-s + 0.742·29-s − 0.176·32-s − 0.696·33-s − 0.342·34-s + 1/6·36-s − 0.493·37-s + 0.162·38-s + 0.160·39-s − 1.87·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−7.85117428893072616672135860245, −6.93464697400332610070867318220, −6.46340726313795801327652856405, −5.35880412768394090518668233654, −4.87492146486785541928812179096, −3.65231316371971387748314334574, −3.03502177507572990369414910309, −2.21586437719846193557180164910, −1.30636718924482078796956899128, 0,
1.30636718924482078796956899128, 2.21586437719846193557180164910, 3.03502177507572990369414910309, 3.65231316371971387748314334574, 4.87492146486785541928812179096, 5.35880412768394090518668233654, 6.46340726313795801327652856405, 6.93464697400332610070867318220, 7.85117428893072616672135860245
