| L(s) = 1 | − 0.611i·3-s + 1.62i·5-s − 3.10·7-s + 2.62·9-s − 5.31i·11-s + i·13-s + 0.994·15-s − 1.89·17-s + 0.885i·19-s + 1.89i·21-s + 1.22·23-s + 2.35·25-s − 3.43i·27-s + 2i·29-s − 3.71·31-s + ⋯ |
| L(s) = 1 | − 0.352i·3-s + 0.727i·5-s − 1.17·7-s + 0.875·9-s − 1.60i·11-s + 0.277i·13-s + 0.256·15-s − 0.460·17-s + 0.203i·19-s + 0.413i·21-s + 0.254·23-s + 0.471·25-s − 0.661i·27-s + 0.371i·29-s − 0.667·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.177720396\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.177720396\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - iT \) |
| good | 3 | \( 1 + 0.611iT - 3T^{2} \) |
| 5 | \( 1 - 1.62iT - 5T^{2} \) |
| 7 | \( 1 + 3.10T + 7T^{2} \) |
| 11 | \( 1 + 5.31iT - 11T^{2} \) |
| 17 | \( 1 + 1.89T + 17T^{2} \) |
| 19 | \( 1 - 0.885iT - 19T^{2} \) |
| 23 | \( 1 - 1.22T + 23T^{2} \) |
| 29 | \( 1 - 2iT - 29T^{2} \) |
| 31 | \( 1 + 3.71T + 31T^{2} \) |
| 37 | \( 1 + 11.1iT - 37T^{2} \) |
| 41 | \( 1 + 5.79T + 41T^{2} \) |
| 43 | \( 1 + 3.43iT - 43T^{2} \) |
| 47 | \( 1 - 0.274T + 47T^{2} \) |
| 53 | \( 1 + 7.79iT - 53T^{2} \) |
| 59 | \( 1 + 1.55iT - 59T^{2} \) |
| 61 | \( 1 + 7.04iT - 61T^{2} \) |
| 67 | \( 1 + 12.7iT - 67T^{2} \) |
| 71 | \( 1 - 8.08T + 71T^{2} \) |
| 73 | \( 1 - 16.2T + 73T^{2} \) |
| 79 | \( 1 - 9.41T + 79T^{2} \) |
| 83 | \( 1 + 10.9iT - 83T^{2} \) |
| 89 | \( 1 + 5.25T + 89T^{2} \) |
| 97 | \( 1 + 8.50T + 97T^{2} \) |
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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
−9.213773987391983728852529518839, −8.402519389876487429708666140523, −7.37272233995363340597252071206, −6.67804491199551521010616887613, −6.24103185771527089784435327603, −5.18310046478905900127216319065, −3.75116060241886287771251299473, −3.27964945092364165152786255033, −2.07518312649043827953340636480, −0.47587281748712482523709718978,
1.29870386067019087911142881534, 2.61180617102227761180491452153, 3.81233516552938289793086496374, 4.60068437689917298588959993927, 5.24250145663347430163459407214, 6.59041637577299919838050528591, 6.98372865832344135354321781756, 7.992582453453163877610791568920, 9.015934554132983029002695712679, 9.657139995092313728210401397617
