| L(s) = 1 | + 3-s − 0.614·5-s − 3.46·7-s + 9-s + 3.80·11-s − 2.93·13-s − 0.614·15-s + 3.15·17-s − 5.58·19-s − 3.46·21-s − 0.574·23-s − 4.62·25-s + 27-s − 3.25·29-s − 1.74·31-s + 3.80·33-s + 2.12·35-s − 4.35·37-s − 2.93·39-s − 1.51·41-s + 9.97·43-s − 0.614·45-s − 8.29·47-s + 4.99·49-s + 3.15·51-s + 0.465·53-s − 2.34·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.275·5-s − 1.30·7-s + 0.333·9-s + 1.14·11-s − 0.812·13-s − 0.158·15-s + 0.765·17-s − 1.28·19-s − 0.755·21-s − 0.119·23-s − 0.924·25-s + 0.192·27-s − 0.603·29-s − 0.313·31-s + 0.663·33-s + 0.359·35-s − 0.716·37-s − 0.469·39-s − 0.236·41-s + 1.52·43-s − 0.0916·45-s − 1.20·47-s + 0.713·49-s + 0.441·51-s + 0.0638·53-s − 0.315·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 + 0.614T + 5T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 11 | \( 1 - 3.80T + 11T^{2} \) |
| 13 | \( 1 + 2.93T + 13T^{2} \) |
| 17 | \( 1 - 3.15T + 17T^{2} \) |
| 19 | \( 1 + 5.58T + 19T^{2} \) |
| 23 | \( 1 + 0.574T + 23T^{2} \) |
| 29 | \( 1 + 3.25T + 29T^{2} \) |
| 31 | \( 1 + 1.74T + 31T^{2} \) |
| 37 | \( 1 + 4.35T + 37T^{2} \) |
| 41 | \( 1 + 1.51T + 41T^{2} \) |
| 43 | \( 1 - 9.97T + 43T^{2} \) |
| 47 | \( 1 + 8.29T + 47T^{2} \) |
| 53 | \( 1 - 0.465T + 53T^{2} \) |
| 59 | \( 1 + 9.58T + 59T^{2} \) |
| 61 | \( 1 + 6.22T + 61T^{2} \) |
| 67 | \( 1 - 3.41T + 67T^{2} \) |
| 71 | \( 1 + 8.58T + 71T^{2} \) |
| 73 | \( 1 + 9.90T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 + 8.39T + 83T^{2} \) |
| 89 | \( 1 + 7.12T + 89T^{2} \) |
| 97 | \( 1 - 4.70T + 97T^{2} \) |
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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
−8.936522961546365081785482276613, −7.968620279335961572443074132100, −7.20052355827743180541626234203, −6.48788355534786335126401042562, −5.74018722167512119730204308119, −4.39497368902159792431797994453, −3.71656765480256856522208934026, −2.91646278961716228305228875553, −1.73789437158128062451700941297, 0,
1.73789437158128062451700941297, 2.91646278961716228305228875553, 3.71656765480256856522208934026, 4.39497368902159792431797994453, 5.74018722167512119730204308119, 6.48788355534786335126401042562, 7.20052355827743180541626234203, 7.968620279335961572443074132100, 8.936522961546365081785482276613
