| L(s) = 1 | + 2.43·3-s − 5-s − 7-s + 2.94·9-s + 1.23·11-s − 13-s − 2.43·15-s − 6.95·17-s + 0.709·19-s − 2.43·21-s + 9.12·23-s + 25-s − 0.133·27-s + 2.66·29-s + 4.30·31-s + 3.01·33-s + 35-s + 3.20·37-s − 2.43·39-s + 5.36·41-s + 9.39·43-s − 2.94·45-s + 11.8·47-s + 49-s − 16.9·51-s + 0.946·53-s − 1.23·55-s + ⋯ |
| L(s) = 1 | + 1.40·3-s − 0.447·5-s − 0.377·7-s + 0.981·9-s + 0.372·11-s − 0.277·13-s − 0.629·15-s − 1.68·17-s + 0.162·19-s − 0.532·21-s + 1.90·23-s + 0.200·25-s − 0.0257·27-s + 0.494·29-s + 0.772·31-s + 0.524·33-s + 0.169·35-s + 0.526·37-s − 0.390·39-s + 0.837·41-s + 1.43·43-s − 0.439·45-s + 1.73·47-s + 0.142·49-s − 2.37·51-s + 0.129·53-s − 0.166·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.797595111\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.797595111\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 2.43T + 3T^{2} \) |
| 11 | \( 1 - 1.23T + 11T^{2} \) |
| 17 | \( 1 + 6.95T + 17T^{2} \) |
| 19 | \( 1 - 0.709T + 19T^{2} \) |
| 23 | \( 1 - 9.12T + 23T^{2} \) |
| 29 | \( 1 - 2.66T + 29T^{2} \) |
| 31 | \( 1 - 4.30T + 31T^{2} \) |
| 37 | \( 1 - 3.20T + 37T^{2} \) |
| 41 | \( 1 - 5.36T + 41T^{2} \) |
| 43 | \( 1 - 9.39T + 43T^{2} \) |
| 47 | \( 1 - 11.8T + 47T^{2} \) |
| 53 | \( 1 - 0.946T + 53T^{2} \) |
| 59 | \( 1 - 9.89T + 59T^{2} \) |
| 61 | \( 1 - 13.9T + 61T^{2} \) |
| 67 | \( 1 + 2.48T + 67T^{2} \) |
| 71 | \( 1 + 0.968T + 71T^{2} \) |
| 73 | \( 1 - 10.0T + 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 + 7.16T + 89T^{2} \) |
| 97 | \( 1 + 8.29T + 97T^{2} \) |
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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
−8.650353457282137807593521195914, −7.930246158459115759604145935867, −7.07714736964018571248121029160, −6.69147413663172540999347234978, −5.47189331573321055478228661813, −4.36134589295590359220716537241, −3.91390611052539731961550035812, −2.76182003493861392257528850689, −2.46409442636902062694617983523, −0.923499141000180457506399484487,
0.923499141000180457506399484487, 2.46409442636902062694617983523, 2.76182003493861392257528850689, 3.91390611052539731961550035812, 4.36134589295590359220716537241, 5.47189331573321055478228661813, 6.69147413663172540999347234978, 7.07714736964018571248121029160, 7.930246158459115759604145935867, 8.650353457282137807593521195914
