| L(s) = 1 | + 1.35·3-s − 5-s − 4.17·7-s − 1.17·9-s + 5.52·11-s − 1.35·15-s − 0.703·17-s − 6.82·19-s − 5.64·21-s − 2.64·23-s + 25-s − 5.64·27-s + 8.17·29-s + 9.52·31-s + 7.46·33-s + 4.17·35-s + 6.87·37-s − 0.703·41-s − 1.35·43-s + 1.17·45-s + 8.17·47-s + 10.4·49-s − 0.951·51-s + 5.04·53-s − 5.52·55-s − 9.22·57-s + 12.2·59-s + ⋯ |
| L(s) = 1 | + 0.780·3-s − 0.447·5-s − 1.57·7-s − 0.390·9-s + 1.66·11-s − 0.349·15-s − 0.170·17-s − 1.56·19-s − 1.23·21-s − 0.552·23-s + 0.200·25-s − 1.08·27-s + 1.51·29-s + 1.71·31-s + 1.30·33-s + 0.705·35-s + 1.13·37-s − 0.109·41-s − 0.206·43-s + 0.174·45-s + 1.19·47-s + 1.48·49-s − 0.133·51-s + 0.693·53-s − 0.744·55-s − 1.22·57-s + 1.59·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.703707426\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.703707426\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 1.35T + 3T^{2} \) |
| 7 | \( 1 + 4.17T + 7T^{2} \) |
| 11 | \( 1 - 5.52T + 11T^{2} \) |
| 17 | \( 1 + 0.703T + 17T^{2} \) |
| 19 | \( 1 + 6.82T + 19T^{2} \) |
| 23 | \( 1 + 2.64T + 23T^{2} \) |
| 29 | \( 1 - 8.17T + 29T^{2} \) |
| 31 | \( 1 - 9.52T + 31T^{2} \) |
| 37 | \( 1 - 6.87T + 37T^{2} \) |
| 41 | \( 1 + 0.703T + 41T^{2} \) |
| 43 | \( 1 + 1.35T + 43T^{2} \) |
| 47 | \( 1 - 8.17T + 47T^{2} \) |
| 53 | \( 1 - 5.04T + 53T^{2} \) |
| 59 | \( 1 - 12.2T + 59T^{2} \) |
| 61 | \( 1 + 0.172T + 61T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 - 5.52T + 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 - 1.29T + 79T^{2} \) |
| 83 | \( 1 - 9.58T + 83T^{2} \) |
| 89 | \( 1 + 11.7T + 89T^{2} \) |
| 97 | \( 1 + 18.3T + 97T^{2} \) |
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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
−8.568079469478263629895008046682, −8.173667052002051144160130316917, −6.90869231036347787382890857156, −6.50032123579405634398691375192, −5.91171560959464989695298085287, −4.34092436391256420509672478062, −3.92374089280354278185285044869, −3.03361506768683978769037860438, −2.32938245388341686087532019388, −0.72689880789068216474831980353,
0.72689880789068216474831980353, 2.32938245388341686087532019388, 3.03361506768683978769037860438, 3.92374089280354278185285044869, 4.34092436391256420509672478062, 5.91171560959464989695298085287, 6.50032123579405634398691375192, 6.90869231036347787382890857156, 8.173667052002051144160130316917, 8.568079469478263629895008046682
