| L(s) = 1 | + 5-s + 2·7-s + 2·11-s + 13-s + 3·17-s + 2·19-s + 6·23-s − 4·25-s + 29-s − 8·31-s + 2·35-s + 37-s − 2·41-s + 10·43-s + 4·47-s − 3·49-s − 10·53-s + 2·55-s + 4·59-s + 9·61-s + 65-s − 14·67-s + 10·71-s − 9·73-s + 4·77-s + 10·79-s + 12·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.755·7-s + 0.603·11-s + 0.277·13-s + 0.727·17-s + 0.458·19-s + 1.25·23-s − 4/5·25-s + 0.185·29-s − 1.43·31-s + 0.338·35-s + 0.164·37-s − 0.312·41-s + 1.52·43-s + 0.583·47-s − 3/7·49-s − 1.37·53-s + 0.269·55-s + 0.520·59-s + 1.15·61-s + 0.124·65-s − 1.71·67-s + 1.18·71-s − 1.05·73-s + 0.455·77-s + 1.12·79-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.422758268\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.422758268\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 9 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−9.087651129989471740169538135388, −8.032336155969553810035891496491, −7.45798600299107264070057160144, −6.56625552207286209190509550821, −5.68540050713803842689031769328, −5.08762444655802351681925910430, −4.07874033083352904439276274319, −3.18931875977735215314606571843, −1.98139167871312679695719285532, −1.06837557476990618375946002698,
1.06837557476990618375946002698, 1.98139167871312679695719285532, 3.18931875977735215314606571843, 4.07874033083352904439276274319, 5.08762444655802351681925910430, 5.68540050713803842689031769328, 6.56625552207286209190509550821, 7.45798600299107264070057160144, 8.032336155969553810035891496491, 9.087651129989471740169538135388
