| L(s) = 1 | − 3-s − 3·5-s − 2·9-s − 4·11-s + 5·13-s + 3·15-s − 3·17-s + 4·19-s + 9·23-s + 4·25-s + 5·27-s + 10·29-s + 4·33-s + 2·37-s − 5·39-s − 3·41-s − 9·43-s + 6·45-s + 3·47-s + 3·51-s − 6·53-s + 12·55-s − 4·57-s − 5·59-s − 7·61-s − 15·65-s − 12·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s − 2/3·9-s − 1.20·11-s + 1.38·13-s + 0.774·15-s − 0.727·17-s + 0.917·19-s + 1.87·23-s + 4/5·25-s + 0.962·27-s + 1.85·29-s + 0.696·33-s + 0.328·37-s − 0.800·39-s − 0.468·41-s − 1.37·43-s + 0.894·45-s + 0.437·47-s + 0.420·51-s − 0.824·53-s + 1.61·55-s − 0.529·57-s − 0.650·59-s − 0.896·61-s − 1.86·65-s − 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 376712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 376712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5886926181\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5886926181\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 31 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 5 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.34281469656456, −11.93080172954957, −11.53145179518337, −11.14314033887311, −10.84397724260436, −10.47446654586291, −9.916512606676211, −9.089779000233739, −8.777013650118962, −8.328140080184116, −8.015494657723961, −7.447325044012212, −6.978237475692930, −6.449251091650980, −6.060028385175891, −5.398138403225646, −4.863862102016510, −4.711348510805958, −3.978952991993061, −3.242981277753846, −3.054998705580384, −2.582396731023465, −1.438042395771919, −0.9709141330105868, −0.2504706666812254,
0.2504706666812254, 0.9709141330105868, 1.438042395771919, 2.582396731023465, 3.054998705580384, 3.242981277753846, 3.978952991993061, 4.711348510805958, 4.863862102016510, 5.398138403225646, 6.060028385175891, 6.449251091650980, 6.978237475692930, 7.447325044012212, 8.015494657723961, 8.328140080184116, 8.777013650118962, 9.089779000233739, 9.916512606676211, 10.47446654586291, 10.84397724260436, 11.14314033887311, 11.53145179518337, 11.93080172954957, 12.34281469656456
