L(s) = 1 | + 5-s − 3.37·7-s − 1.37·11-s + 13-s + 7.37·17-s + 4·19-s − 7.37·23-s + 25-s − 2.74·29-s + 6.74·31-s − 3.37·35-s − 8.11·37-s − 1.37·41-s − 10.7·43-s + 11.4·47-s + 4.37·49-s + 7.37·53-s − 1.37·55-s − 8.74·59-s + 6.62·61-s + 65-s + 4·67-s − 1.37·71-s + 7.48·73-s + 4.62·77-s + 16.8·79-s − 3.25·83-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 1.27·7-s − 0.413·11-s + 0.277·13-s + 1.78·17-s + 0.917·19-s − 1.53·23-s + 0.200·25-s − 0.509·29-s + 1.21·31-s − 0.570·35-s − 1.33·37-s − 0.214·41-s − 1.63·43-s + 1.67·47-s + 0.624·49-s + 1.01·53-s − 0.185·55-s − 1.13·59-s + 0.848·61-s + 0.124·65-s + 0.488·67-s − 0.162·71-s + 0.876·73-s + 0.527·77-s + 1.89·79-s − 0.357·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.753873871\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.753873871\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + 3.37T + 7T^{2} \) |
| 11 | \( 1 + 1.37T + 11T^{2} \) |
| 17 | \( 1 - 7.37T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 7.37T + 23T^{2} \) |
| 29 | \( 1 + 2.74T + 29T^{2} \) |
| 31 | \( 1 - 6.74T + 31T^{2} \) |
| 37 | \( 1 + 8.11T + 37T^{2} \) |
| 41 | \( 1 + 1.37T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 - 7.37T + 53T^{2} \) |
| 59 | \( 1 + 8.74T + 59T^{2} \) |
| 61 | \( 1 - 6.62T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 1.37T + 71T^{2} \) |
| 73 | \( 1 - 7.48T + 73T^{2} \) |
| 79 | \( 1 - 16.8T + 79T^{2} \) |
| 83 | \( 1 + 3.25T + 83T^{2} \) |
| 89 | \( 1 + 4.11T + 89T^{2} \) |
| 97 | \( 1 + 17.3T + 97T^{2} \) |
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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
−7.77101989528550341770047707732, −6.91141796006616364677409476381, −6.35851277295490706870064120922, −5.55954538757103709283753768432, −5.28743321405102824054128997708, −3.98890704159087690979832426438, −3.39344704474328219154890807134, −2.76975286392117622157820202433, −1.71926944566319832974660438042, −0.63433802280451236892629522708,
0.63433802280451236892629522708, 1.71926944566319832974660438042, 2.76975286392117622157820202433, 3.39344704474328219154890807134, 3.98890704159087690979832426438, 5.28743321405102824054128997708, 5.55954538757103709283753768432, 6.35851277295490706870064120922, 6.91141796006616364677409476381, 7.77101989528550341770047707732