| L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s + 5·5-s + 6·6-s + 7·7-s + 8·8-s + 9·9-s + 10·10-s + 11·11-s + 12·12-s − 28·13-s + 14·14-s + 15·15-s + 16·16-s − 79·17-s + 18·18-s − 75·19-s + 20·20-s + 21·21-s + 22·22-s − 183·23-s + 24·24-s + 25·25-s − 56·26-s + 27·27-s + 28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s + 0.288·12-s − 0.597·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 1.12·17-s + 0.235·18-s − 0.905·19-s + 0.223·20-s + 0.218·21-s + 0.213·22-s − 1.65·23-s + 0.204·24-s + 1/5·25-s − 0.422·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2310 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 - p T \) |
| 11 | \( 1 - p T \) |
| good | 13 | \( 1 + 28 T + p^{3} T^{2} \) |
| 17 | \( 1 + 79 T + p^{3} T^{2} \) |
| 19 | \( 1 + 75 T + p^{3} T^{2} \) |
| 23 | \( 1 + 183 T + p^{3} T^{2} \) |
| 29 | \( 1 + 215 T + p^{3} T^{2} \) |
| 31 | \( 1 + 198 T + p^{3} T^{2} \) |
| 37 | \( 1 + 64 T + p^{3} T^{2} \) |
| 41 | \( 1 + 378 T + p^{3} T^{2} \) |
| 43 | \( 1 - 37 T + p^{3} T^{2} \) |
| 47 | \( 1 + 334 T + p^{3} T^{2} \) |
| 53 | \( 1 + 183 T + p^{3} T^{2} \) |
| 59 | \( 1 - 495 T + p^{3} T^{2} \) |
| 61 | \( 1 - 527 T + p^{3} T^{2} \) |
| 67 | \( 1 + 554 T + p^{3} T^{2} \) |
| 71 | \( 1 + 138 T + p^{3} T^{2} \) |
| 73 | \( 1 - 12 T + p^{3} T^{2} \) |
| 79 | \( 1 - 160 T + p^{3} T^{2} \) |
| 83 | \( 1 - 357 T + p^{3} T^{2} \) |
| 89 | \( 1 + 355 T + p^{3} T^{2} \) |
| 97 | \( 1 - 721 T + p^{3} T^{2} \) |
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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
−8.275643169265510835007690527340, −7.39776329175911896378766732700, −6.67396810324890583374610104210, −5.88189551988311380342814989498, −4.98036938452911164153267378296, −4.18491397100696046281714373868, −3.44372680461899829906279711068, −2.09024866066738387390394741378, −1.90724541481367317219268621605, 0,
1.90724541481367317219268621605, 2.09024866066738387390394741378, 3.44372680461899829906279711068, 4.18491397100696046281714373868, 4.98036938452911164153267378296, 5.88189551988311380342814989498, 6.67396810324890583374610104210, 7.39776329175911896378766732700, 8.275643169265510835007690527340
