| L(s) = 1 | − 2.27·3-s − 5-s + 7-s + 2.17·9-s − 3.56·11-s + 13-s + 2.27·15-s − 1.60·17-s + 4.94·19-s − 2.27·21-s − 1.01·23-s + 25-s + 1.87·27-s + 1.84·29-s − 5.83·31-s + 8.11·33-s − 35-s + 9.41·37-s − 2.27·39-s − 1.41·41-s − 3.75·43-s − 2.17·45-s − 2.01·47-s + 49-s + 3.64·51-s − 2.58·53-s + 3.56·55-s + ⋯ |
| L(s) = 1 | − 1.31·3-s − 0.447·5-s + 0.377·7-s + 0.724·9-s − 1.07·11-s + 0.277·13-s + 0.587·15-s − 0.389·17-s + 1.13·19-s − 0.496·21-s − 0.212·23-s + 0.200·25-s + 0.361·27-s + 0.342·29-s − 1.04·31-s + 1.41·33-s − 0.169·35-s + 1.54·37-s − 0.364·39-s − 0.221·41-s − 0.573·43-s − 0.324·45-s − 0.293·47-s + 0.142·49-s + 0.511·51-s − 0.355·53-s + 0.481·55-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.27T + 3T^{2} \) |
| 11 | \( 1 + 3.56T + 11T^{2} \) |
| 17 | \( 1 + 1.60T + 17T^{2} \) |
| 19 | \( 1 - 4.94T + 19T^{2} \) |
| 23 | \( 1 + 1.01T + 23T^{2} \) |
| 29 | \( 1 - 1.84T + 29T^{2} \) |
| 31 | \( 1 + 5.83T + 31T^{2} \) |
| 37 | \( 1 - 9.41T + 37T^{2} \) |
| 41 | \( 1 + 1.41T + 41T^{2} \) |
| 43 | \( 1 + 3.75T + 43T^{2} \) |
| 47 | \( 1 + 2.01T + 47T^{2} \) |
| 53 | \( 1 + 2.58T + 53T^{2} \) |
| 59 | \( 1 - 4.81T + 59T^{2} \) |
| 61 | \( 1 - 8.53T + 61T^{2} \) |
| 67 | \( 1 - 1.41T + 67T^{2} \) |
| 71 | \( 1 + 8.75T + 71T^{2} \) |
| 73 | \( 1 - 0.229T + 73T^{2} \) |
| 79 | \( 1 - 5.96T + 79T^{2} \) |
| 83 | \( 1 + 8.54T + 83T^{2} \) |
| 89 | \( 1 - 10.1T + 89T^{2} \) |
| 97 | \( 1 - 7.67T + 97T^{2} \) |
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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
−8.009174488694886823090397828995, −7.43015403593904368780204234587, −6.60772546044935279550151214018, −5.79840060996808839473112149228, −5.19230276214709450007200196497, −4.61268724320926132957321139104, −3.56912539047352064129535037306, −2.48748319785975084713221044228, −1.11731396814719102566903585401, 0,
1.11731396814719102566903585401, 2.48748319785975084713221044228, 3.56912539047352064129535037306, 4.61268724320926132957321139104, 5.19230276214709450007200196497, 5.79840060996808839473112149228, 6.60772546044935279550151214018, 7.43015403593904368780204234587, 8.009174488694886823090397828995
