| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 6·6-s + 15·7-s + 8·8-s + 9·9-s + 39·11-s − 12·12-s + 13·13-s + 30·14-s + 16·16-s + 15·17-s + 18·18-s + 54·19-s − 45·21-s + 78·22-s + 143·23-s − 24·24-s + 26·26-s − 27·27-s + 60·28-s − 122·29-s − 246·31-s + 32·32-s − 117·33-s + 30·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.809·7-s + 0.353·8-s + 1/3·9-s + 1.06·11-s − 0.288·12-s + 0.277·13-s + 0.572·14-s + 1/4·16-s + 0.214·17-s + 0.235·18-s + 0.652·19-s − 0.467·21-s + 0.755·22-s + 1.29·23-s − 0.204·24-s + 0.196·26-s − 0.192·27-s + 0.404·28-s − 0.781·29-s − 1.42·31-s + 0.176·32-s − 0.617·33-s + 0.151·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.986502911\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.986502911\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - p T \) |
| good | 7 | \( 1 - 15 T + p^{3} T^{2} \) |
| 11 | \( 1 - 39 T + p^{3} T^{2} \) |
| 17 | \( 1 - 15 T + p^{3} T^{2} \) |
| 19 | \( 1 - 54 T + p^{3} T^{2} \) |
| 23 | \( 1 - 143 T + p^{3} T^{2} \) |
| 29 | \( 1 + 122 T + p^{3} T^{2} \) |
| 31 | \( 1 + 246 T + p^{3} T^{2} \) |
| 37 | \( 1 - 225 T + p^{3} T^{2} \) |
| 41 | \( 1 - 469 T + p^{3} T^{2} \) |
| 43 | \( 1 - 484 T + p^{3} T^{2} \) |
| 47 | \( 1 + 234 T + p^{3} T^{2} \) |
| 53 | \( 1 + 33 T + p^{3} T^{2} \) |
| 59 | \( 1 + p^{3} T^{2} \) |
| 61 | \( 1 + 831 T + p^{3} T^{2} \) |
| 67 | \( 1 + 772 T + p^{3} T^{2} \) |
| 71 | \( 1 + 793 T + p^{3} T^{2} \) |
| 73 | \( 1 - 998 T + p^{3} T^{2} \) |
| 79 | \( 1 + 681 T + p^{3} T^{2} \) |
| 83 | \( 1 - 772 T + p^{3} T^{2} \) |
| 89 | \( 1 + 465 T + p^{3} T^{2} \) |
| 97 | \( 1 - 79 T + p^{3} T^{2} \) |
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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
−9.004358689342974940978557291478, −7.68374103747363030864278614455, −7.28224049035386900124633565358, −6.20342674204878330958591318274, −5.64802109281448721165959350191, −4.74440673759199361811606982539, −4.07499490139641296320522557777, −3.07895990596581502817824740128, −1.74180837154078143913165407457, −0.927524711179559348904529451484,
0.927524711179559348904529451484, 1.74180837154078143913165407457, 3.07895990596581502817824740128, 4.07499490139641296320522557777, 4.74440673759199361811606982539, 5.64802109281448721165959350191, 6.20342674204878330958591318274, 7.28224049035386900124633565358, 7.68374103747363030864278614455, 9.004358689342974940978557291478
