| L(s) = 1 | − 3-s − 2.41·5-s + 4.14·7-s + 9-s + 3.46·11-s + 2.41·15-s + 6.17·17-s − 3.46·19-s − 4.14·21-s − 2·23-s + 0.821·25-s − 27-s − 8.17·29-s − 7.60·31-s − 3.46·33-s − 10·35-s − 1.05·37-s + 5.87·41-s − 0.821·43-s − 2.41·45-s − 10.3·47-s + 10.1·49-s − 6.17·51-s − 10.1·53-s − 8.35·55-s + 3.46·57-s − 1.36·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.07·5-s + 1.56·7-s + 0.333·9-s + 1.04·11-s + 0.622·15-s + 1.49·17-s − 0.794·19-s − 0.904·21-s − 0.417·23-s + 0.164·25-s − 0.192·27-s − 1.51·29-s − 1.36·31-s − 0.603·33-s − 1.69·35-s − 0.172·37-s + 0.917·41-s − 0.125·43-s − 0.359·45-s − 1.51·47-s + 1.45·49-s − 0.865·51-s − 1.39·53-s − 1.12·55-s + 0.458·57-s − 0.177·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 2.41T + 5T^{2} \) |
| 7 | \( 1 - 4.14T + 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 17 | \( 1 - 6.17T + 17T^{2} \) |
| 19 | \( 1 + 3.46T + 19T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 + 8.17T + 29T^{2} \) |
| 31 | \( 1 + 7.60T + 31T^{2} \) |
| 37 | \( 1 + 1.05T + 37T^{2} \) |
| 41 | \( 1 - 5.87T + 41T^{2} \) |
| 43 | \( 1 + 0.821T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 + 1.36T + 59T^{2} \) |
| 61 | \( 1 - 5T + 61T^{2} \) |
| 67 | \( 1 + 4.14T + 67T^{2} \) |
| 71 | \( 1 + 3.46T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 + 13.1T + 79T^{2} \) |
| 83 | \( 1 + 11.7T + 83T^{2} \) |
| 89 | \( 1 + 6.92T + 89T^{2} \) |
| 97 | \( 1 - 2.04T + 97T^{2} \) |
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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
−7.63262689676759502785806712040, −6.94073172273893488171980187132, −5.95629502155798144366549362561, −5.39243743965939358812305194602, −4.57735376555781538014683604923, −4.00634071939119212588721287595, −3.40639691840839961577168402403, −1.88637684287730260250900710132, −1.30293239530678330219166392615, 0,
1.30293239530678330219166392615, 1.88637684287730260250900710132, 3.40639691840839961577168402403, 4.00634071939119212588721287595, 4.57735376555781538014683604923, 5.39243743965939358812305194602, 5.95629502155798144366549362561, 6.94073172273893488171980187132, 7.63262689676759502785806712040
