| L(s) = 1 | − 3-s + 5-s − 1.61·7-s + 9-s − 4.23·11-s − 13-s − 15-s + 3.85·17-s + 5.85·19-s + 1.61·21-s + 1.85·23-s − 4·25-s − 27-s + 3.61·29-s − 2.85·31-s + 4.23·33-s − 1.61·35-s − 3.38·37-s + 39-s − 6.09·41-s + 9.94·43-s + 45-s − 4.38·47-s − 4.38·49-s − 3.85·51-s − 9.94·53-s − 4.23·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.611·7-s + 0.333·9-s − 1.27·11-s − 0.277·13-s − 0.258·15-s + 0.934·17-s + 1.34·19-s + 0.353·21-s + 0.386·23-s − 0.800·25-s − 0.192·27-s + 0.671·29-s − 0.512·31-s + 0.737·33-s − 0.273·35-s − 0.555·37-s + 0.160·39-s − 0.951·41-s + 1.51·43-s + 0.149·45-s − 0.639·47-s − 0.625·49-s − 0.539·51-s − 1.36·53-s − 0.571·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| good | 5 | \( 1 - T + 5T^{2} \) |
| 7 | \( 1 + 1.61T + 7T^{2} \) |
| 11 | \( 1 + 4.23T + 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 - 3.85T + 17T^{2} \) |
| 19 | \( 1 - 5.85T + 19T^{2} \) |
| 23 | \( 1 - 1.85T + 23T^{2} \) |
| 29 | \( 1 - 3.61T + 29T^{2} \) |
| 31 | \( 1 + 2.85T + 31T^{2} \) |
| 37 | \( 1 + 3.38T + 37T^{2} \) |
| 41 | \( 1 + 6.09T + 41T^{2} \) |
| 43 | \( 1 - 9.94T + 43T^{2} \) |
| 47 | \( 1 + 4.38T + 47T^{2} \) |
| 53 | \( 1 + 9.94T + 53T^{2} \) |
| 61 | \( 1 - 2.85T + 61T^{2} \) |
| 67 | \( 1 + 11.9T + 67T^{2} \) |
| 71 | \( 1 - 4.70T + 71T^{2} \) |
| 73 | \( 1 - 0.381T + 73T^{2} \) |
| 79 | \( 1 + 13.9T + 79T^{2} \) |
| 83 | \( 1 + 3.14T + 83T^{2} \) |
| 89 | \( 1 + 8.09T + 89T^{2} \) |
| 97 | \( 1 + 8.70T + 97T^{2} \) |
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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
−8.297383721934684444907930520465, −7.56355736386737833666702231336, −6.93272019162087003673833162634, −5.89997687435219092989708057310, −5.45542118668332726582780909830, −4.71944909123411636221546882965, −3.43411685479387234769702805391, −2.71779722904219588949226758799, −1.40697437058317306517107496252, 0,
1.40697437058317306517107496252, 2.71779722904219588949226758799, 3.43411685479387234769702805391, 4.71944909123411636221546882965, 5.45542118668332726582780909830, 5.89997687435219092989708057310, 6.93272019162087003673833162634, 7.56355736386737833666702231336, 8.297383721934684444907930520465
