| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s − 4·11-s + 12-s + 4·13-s + 16-s − 6·17-s − 18-s − 19-s + 4·22-s − 24-s − 4·26-s + 27-s − 2·29-s − 32-s − 4·33-s + 6·34-s + 36-s + 4·37-s + 38-s + 4·39-s − 12·41-s − 6·43-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 + 1.10·13-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.229·19-s + 0.852·22-s − 0.204·24-s − 0.784·26-s + 0.192·27-s − 0.371·29-s − 0.176·32-s − 0.696·33-s + 1.02·34-s + 1/6·36-s + 0.657·37-s + 0.162·38-s + 0.640·39-s − 1.87·41-s − 0.914·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 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
−8.476436746654235442316290282700, −7.900013935762179774135988802550, −7.00034238848896644633339370068, −6.35464316636008234486207761930, −5.37040663084008674926727251196, −4.38171405237765230161352358718, −3.36875425082195307584132862105, −2.48339574450153997622074091080, −1.58016455206581811988513488794, 0,
1.58016455206581811988513488794, 2.48339574450153997622074091080, 3.36875425082195307584132862105, 4.38171405237765230161352358718, 5.37040663084008674926727251196, 6.35464316636008234486207761930, 7.00034238848896644633339370068, 7.900013935762179774135988802550, 8.476436746654235442316290282700
