| L(s) = 1 | − 3·3-s + 6·5-s + 4·7-s + 9·9-s − 36·11-s − 18·15-s + 66·17-s − 56·19-s − 12·21-s + 96·23-s − 89·25-s − 27·27-s + 222·29-s − 260·31-s + 108·33-s + 24·35-s + 106·37-s + 90·41-s + 44·43-s + 54·45-s − 168·47-s − 327·49-s − 198·51-s + 30·53-s − 216·55-s + 168·57-s − 348·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.536·5-s + 0.215·7-s + 1/3·9-s − 0.986·11-s − 0.309·15-s + 0.941·17-s − 0.676·19-s − 0.124·21-s + 0.870·23-s − 0.711·25-s − 0.192·27-s + 1.42·29-s − 1.50·31-s + 0.569·33-s + 0.115·35-s + 0.470·37-s + 0.342·41-s + 0.156·43-s + 0.178·45-s − 0.521·47-s − 0.953·49-s − 0.543·51-s + 0.0777·53-s − 0.529·55-s + 0.390·57-s − 0.767·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 6 T + p^{3} T^{2} \) |
| 7 | \( 1 - 4 T + p^{3} T^{2} \) |
| 11 | \( 1 + 36 T + p^{3} T^{2} \) |
| 17 | \( 1 - 66 T + p^{3} T^{2} \) |
| 19 | \( 1 + 56 T + p^{3} T^{2} \) |
| 23 | \( 1 - 96 T + p^{3} T^{2} \) |
| 29 | \( 1 - 222 T + p^{3} T^{2} \) |
| 31 | \( 1 + 260 T + p^{3} T^{2} \) |
| 37 | \( 1 - 106 T + p^{3} T^{2} \) |
| 41 | \( 1 - 90 T + p^{3} T^{2} \) |
| 43 | \( 1 - 44 T + p^{3} T^{2} \) |
| 47 | \( 1 + 168 T + p^{3} T^{2} \) |
| 53 | \( 1 - 30 T + p^{3} T^{2} \) |
| 59 | \( 1 + 348 T + p^{3} T^{2} \) |
| 61 | \( 1 + 346 T + p^{3} T^{2} \) |
| 67 | \( 1 - 256 T + p^{3} T^{2} \) |
| 71 | \( 1 - 168 T + p^{3} T^{2} \) |
| 73 | \( 1 - 814 T + p^{3} T^{2} \) |
| 79 | \( 1 - 200 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1236 T + p^{3} T^{2} \) |
| 89 | \( 1 + 318 T + p^{3} T^{2} \) |
| 97 | \( 1 - 502 T + p^{3} T^{2} \) |
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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
−8.283189583478755422864879115629, −7.64880575573932426851964815984, −6.73968272181127938706529291015, −5.90215757044696744174030237577, −5.27190368082399333646173217022, −4.52805242521741617109923155827, −3.32182419238289779204721457512, −2.28913875228638681264866107143, −1.22574378422677359269011871093, 0,
1.22574378422677359269011871093, 2.28913875228638681264866107143, 3.32182419238289779204721457512, 4.52805242521741617109923155827, 5.27190368082399333646173217022, 5.90215757044696744174030237577, 6.73968272181127938706529291015, 7.64880575573932426851964815984, 8.283189583478755422864879115629
