| L(s) = 1 | + 0.837·3-s + 3.44·5-s − 0.535·7-s − 2.29·9-s − 5.54·13-s + 2.88·15-s − 1.49·17-s + 19-s − 0.448·21-s + 3.26·23-s + 6.90·25-s − 4.43·27-s + 3.17·29-s − 0.888·31-s − 1.84·35-s + 11.0·37-s − 4.64·39-s + 8.44·41-s + 7.61·43-s − 7.93·45-s + 3.39·47-s − 6.71·49-s − 1.25·51-s + 7.22·53-s + 0.837·57-s − 1.36·59-s + 2.57·61-s + ⋯ |
| L(s) = 1 | + 0.483·3-s + 1.54·5-s − 0.202·7-s − 0.766·9-s − 1.53·13-s + 0.745·15-s − 0.363·17-s + 0.229·19-s − 0.0979·21-s + 0.680·23-s + 1.38·25-s − 0.853·27-s + 0.589·29-s − 0.159·31-s − 0.312·35-s + 1.81·37-s − 0.743·39-s + 1.31·41-s + 1.16·43-s − 1.18·45-s + 0.495·47-s − 0.958·49-s − 0.175·51-s + 0.992·53-s + 0.110·57-s − 0.177·59-s + 0.329·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.875273369\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.875273369\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 0.837T + 3T^{2} \) |
| 5 | \( 1 - 3.44T + 5T^{2} \) |
| 7 | \( 1 + 0.535T + 7T^{2} \) |
| 13 | \( 1 + 5.54T + 13T^{2} \) |
| 17 | \( 1 + 1.49T + 17T^{2} \) |
| 23 | \( 1 - 3.26T + 23T^{2} \) |
| 29 | \( 1 - 3.17T + 29T^{2} \) |
| 31 | \( 1 + 0.888T + 31T^{2} \) |
| 37 | \( 1 - 11.0T + 37T^{2} \) |
| 41 | \( 1 - 8.44T + 41T^{2} \) |
| 43 | \( 1 - 7.61T + 43T^{2} \) |
| 47 | \( 1 - 3.39T + 47T^{2} \) |
| 53 | \( 1 - 7.22T + 53T^{2} \) |
| 59 | \( 1 + 1.36T + 59T^{2} \) |
| 61 | \( 1 - 2.57T + 61T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 - 2.66T + 71T^{2} \) |
| 73 | \( 1 + 9.82T + 73T^{2} \) |
| 79 | \( 1 - 2.17T + 79T^{2} \) |
| 83 | \( 1 + 4.71T + 83T^{2} \) |
| 89 | \( 1 + 4.66T + 89T^{2} \) |
| 97 | \( 1 - 6.93T + 97T^{2} \) |
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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
−7.58134177153483323564508807930, −7.15886875082479425847538405599, −6.05522588607938688883895761254, −5.90567766226701141140122285506, −4.98846979934525950694805671284, −4.37299538420998782148306500062, −3.06546868282280441265336217317, −2.57492360677729809891743874544, −2.04676837653148180190298256077, −0.77351046281811203432085666999,
0.77351046281811203432085666999, 2.04676837653148180190298256077, 2.57492360677729809891743874544, 3.06546868282280441265336217317, 4.37299538420998782148306500062, 4.98846979934525950694805671284, 5.90567766226701141140122285506, 6.05522588607938688883895761254, 7.15886875082479425847538405599, 7.58134177153483323564508807930
