| L(s) = 1 | − 2·5-s + 4·13-s − 8·17-s − 25-s + 10·29-s + 12·37-s + 8·41-s − 7·49-s − 14·53-s + 12·61-s − 8·65-s + 6·73-s + 16·85-s + 16·89-s + 18·97-s + 2·101-s + 20·109-s + 16·113-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.10·13-s − 1.94·17-s − 1/5·25-s + 1.85·29-s + 1.97·37-s + 1.24·41-s − 49-s − 1.92·53-s + 1.53·61-s − 0.992·65-s + 0.702·73-s + 1.73·85-s + 1.69·89-s + 1.82·97-s + 0.199·101-s + 1.91·109-s + 1.50·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.367913948\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.367913948\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−8.861792168523134861661213935674, −8.242790060565164263842729426674, −7.61265699519221135849619148511, −6.51852827413755820930431404947, −6.17025673306416658835216275747, −4.73273267023267405907317724557, −4.25715037631953280136020371932, −3.30329965092489688314729963431, −2.22519740698268929137377439052, −0.75200942853633275817465282286,
0.75200942853633275817465282286, 2.22519740698268929137377439052, 3.30329965092489688314729963431, 4.25715037631953280136020371932, 4.73273267023267405907317724557, 6.17025673306416658835216275747, 6.51852827413755820930431404947, 7.61265699519221135849619148511, 8.242790060565164263842729426674, 8.861792168523134861661213935674
