| L(s) = 1 | − 3·5-s + 2·7-s − 11-s + 13-s + 2·19-s + 3·23-s + 4·25-s + 6·29-s − 31-s − 6·35-s − 7·37-s − 6·41-s + 8·43-s − 12·47-s − 3·49-s + 6·53-s + 3·55-s − 9·59-s + 2·61-s − 3·65-s − 7·67-s + 3·71-s + 8·73-s − 2·77-s − 4·79-s + 12·83-s + 15·89-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.755·7-s − 0.301·11-s + 0.277·13-s + 0.458·19-s + 0.625·23-s + 4/5·25-s + 1.11·29-s − 0.179·31-s − 1.01·35-s − 1.15·37-s − 0.937·41-s + 1.21·43-s − 1.75·47-s − 3/7·49-s + 0.824·53-s + 0.404·55-s − 1.17·59-s + 0.256·61-s − 0.372·65-s − 0.855·67-s + 0.356·71-s + 0.936·73-s − 0.227·77-s − 0.450·79-s + 1.31·83-s + 1.58·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.447335025\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.447335025\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−8.239332110498519908064819178334, −7.55418408665461619946322785127, −7.00405918077482307851361858203, −6.09288236530196678208973484981, −5.00121141080351479558787685043, −4.67206110909396431054061165312, −3.65302000242077202036089316902, −3.07735109747862345434472923122, −1.81209257581492259318411550951, −0.66159197553729553396209728874,
0.66159197553729553396209728874, 1.81209257581492259318411550951, 3.07735109747862345434472923122, 3.65302000242077202036089316902, 4.67206110909396431054061165312, 5.00121141080351479558787685043, 6.09288236530196678208973484981, 7.00405918077482307851361858203, 7.55418408665461619946322785127, 8.239332110498519908064819178334
