| L(s) = 1 | + 3-s + 3.23·5-s − 1.23·7-s + 9-s − 4·11-s − 4.47·13-s + 3.23·15-s − 7.23·17-s − 2.76·19-s − 1.23·21-s + 23-s + 5.47·25-s + 27-s + 4.47·29-s + 2.47·31-s − 4·33-s − 4.00·35-s + 4.47·37-s − 4.47·39-s + 6.94·41-s − 7.70·43-s + 3.23·45-s − 4·47-s − 5.47·49-s − 7.23·51-s + 0.763·53-s − 12.9·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.44·5-s − 0.467·7-s + 0.333·9-s − 1.20·11-s − 1.24·13-s + 0.835·15-s − 1.75·17-s − 0.634·19-s − 0.269·21-s + 0.208·23-s + 1.09·25-s + 0.192·27-s + 0.830·29-s + 0.444·31-s − 0.696·33-s − 0.676·35-s + 0.735·37-s − 0.716·39-s + 1.08·41-s − 1.17·43-s + 0.482·45-s − 0.583·47-s − 0.781·49-s − 1.01·51-s + 0.104·53-s − 1.74·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - 3.23T + 5T^{2} \) |
| 7 | \( 1 + 1.23T + 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 + 7.23T + 17T^{2} \) |
| 19 | \( 1 + 2.76T + 19T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 - 2.47T + 31T^{2} \) |
| 37 | \( 1 - 4.47T + 37T^{2} \) |
| 41 | \( 1 - 6.94T + 41T^{2} \) |
| 43 | \( 1 + 7.70T + 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 - 0.763T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 - 4.47T + 61T^{2} \) |
| 67 | \( 1 + 5.23T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 10.9T + 73T^{2} \) |
| 79 | \( 1 + 3.70T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 - 3.23T + 89T^{2} \) |
| 97 | \( 1 + 0.472T + 97T^{2} \) |
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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
−8.084051926373038396915497381252, −7.20168892735224606665913517515, −6.51165796321878820127652972834, −5.90514808461132253692573244477, −4.89226018923646975551211011261, −4.45972247054425445286630527945, −2.88091064853498118821036685962, −2.55691344686327950583785915245, −1.75020256190796892069954039788, 0,
1.75020256190796892069954039788, 2.55691344686327950583785915245, 2.88091064853498118821036685962, 4.45972247054425445286630527945, 4.89226018923646975551211011261, 5.90514808461132253692573244477, 6.51165796321878820127652972834, 7.20168892735224606665913517515, 8.084051926373038396915497381252
