L(s) = 1 | + 5-s + 4.82·7-s − 11-s + 5.65·13-s + 6.82·17-s + 1.17·19-s − 4·23-s + 25-s − 0.828·29-s + 4.82·35-s + 0.343·37-s + 0.828·41-s + 3.17·43-s − 4·47-s + 16.3·49-s + 13.3·53-s − 55-s − 4·59-s − 0.343·61-s + 5.65·65-s − 5.65·67-s + 13.6·71-s − 11.3·73-s − 4.82·77-s + 8.48·79-s − 10·83-s + 6.82·85-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 1.82·7-s − 0.301·11-s + 1.56·13-s + 1.65·17-s + 0.268·19-s − 0.834·23-s + 0.200·25-s − 0.153·29-s + 0.816·35-s + 0.0564·37-s + 0.129·41-s + 0.483·43-s − 0.583·47-s + 2.33·49-s + 1.82·53-s − 0.134·55-s − 0.520·59-s − 0.0439·61-s + 0.701·65-s − 0.691·67-s + 1.62·71-s − 1.32·73-s − 0.550·77-s + 0.954·79-s − 1.09·83-s + 0.740·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.521726498\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.521726498\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 7 | \( 1 - 4.82T + 7T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 17 | \( 1 - 6.82T + 17T^{2} \) |
| 19 | \( 1 - 1.17T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 0.828T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 0.343T + 37T^{2} \) |
| 41 | \( 1 - 0.828T + 41T^{2} \) |
| 43 | \( 1 - 3.17T + 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 0.343T + 61T^{2} \) |
| 67 | \( 1 + 5.65T + 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 - 8.48T + 79T^{2} \) |
| 83 | \( 1 + 10T + 83T^{2} \) |
| 89 | \( 1 - 7.65T + 89T^{2} \) |
| 97 | \( 1 - 0.343T + 97T^{2} \) |
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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
−7.965381801156168283272471023746, −7.36095641566394068722333768797, −6.31981750987947601465700547473, −5.55527703368403334649323090761, −5.28952618023231337302848626887, −4.26923387341740387707431209998, −3.63495828976766875045445310362, −2.56074077660263243698300210894, −1.57212968246214759875529195810, −1.08216848792811576668245872007,
1.08216848792811576668245872007, 1.57212968246214759875529195810, 2.56074077660263243698300210894, 3.63495828976766875045445310362, 4.26923387341740387707431209998, 5.28952618023231337302848626887, 5.55527703368403334649323090761, 6.31981750987947601465700547473, 7.36095641566394068722333768797, 7.965381801156168283272471023746