| L(s) = 1 | + 5-s + 2.73·7-s − 5.52·11-s − 3.52·13-s + 5.52·17-s + 7.52·19-s − 0.734·23-s + 25-s + 4.46·29-s + 7.52·31-s + 2.73·35-s + 6.05·37-s + 1.05·41-s − 3.52·43-s − 1.20·47-s + 0.475·49-s − 1.46·53-s − 5.52·55-s + 1.46·59-s + 9.05·61-s − 3.52·65-s − 4.25·67-s − 10.0·71-s + 8·73-s − 15.1·77-s + 2·79-s + 5.26·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.03·7-s − 1.66·11-s − 0.977·13-s + 1.33·17-s + 1.72·19-s − 0.153·23-s + 0.200·25-s + 0.829·29-s + 1.35·31-s + 0.462·35-s + 0.995·37-s + 0.164·41-s − 0.537·43-s − 0.176·47-s + 0.0679·49-s − 0.201·53-s − 0.744·55-s + 0.191·59-s + 1.15·61-s − 0.437·65-s − 0.520·67-s − 1.19·71-s + 0.936·73-s − 1.72·77-s + 0.225·79-s + 0.577·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.999582228\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.999582228\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 7 | \( 1 - 2.73T + 7T^{2} \) |
| 11 | \( 1 + 5.52T + 11T^{2} \) |
| 13 | \( 1 + 3.52T + 13T^{2} \) |
| 17 | \( 1 - 5.52T + 17T^{2} \) |
| 19 | \( 1 - 7.52T + 19T^{2} \) |
| 23 | \( 1 + 0.734T + 23T^{2} \) |
| 29 | \( 1 - 4.46T + 29T^{2} \) |
| 31 | \( 1 - 7.52T + 31T^{2} \) |
| 37 | \( 1 - 6.05T + 37T^{2} \) |
| 41 | \( 1 - 1.05T + 41T^{2} \) |
| 43 | \( 1 + 3.52T + 43T^{2} \) |
| 47 | \( 1 + 1.20T + 47T^{2} \) |
| 53 | \( 1 + 1.46T + 53T^{2} \) |
| 59 | \( 1 - 1.46T + 59T^{2} \) |
| 61 | \( 1 - 9.05T + 61T^{2} \) |
| 67 | \( 1 + 4.25T + 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 - 8T + 73T^{2} \) |
| 79 | \( 1 - 2T + 79T^{2} \) |
| 83 | \( 1 - 5.26T + 83T^{2} \) |
| 89 | \( 1 - 3T + 89T^{2} \) |
| 97 | \( 1 - 9.46T + 97T^{2} \) |
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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
−9.736639289727599557711373686456, −8.342204531456090311354677390391, −7.83655643456665478985981197567, −7.26594751963026431141850462482, −5.93952620643909109879999078487, −5.13382741899647136266343243336, −4.78104532984717286088583198403, −3.15666010952137241106867011650, −2.39045254788942127022351132667, −1.03856821501147150125218615389,
1.03856821501147150125218615389, 2.39045254788942127022351132667, 3.15666010952137241106867011650, 4.78104532984717286088583198403, 5.13382741899647136266343243336, 5.93952620643909109879999078487, 7.26594751963026431141850462482, 7.83655643456665478985981197567, 8.342204531456090311354677390391, 9.736639289727599557711373686456
