| L(s) = 1 | + 2.56·3-s + 7-s + 3.56·9-s − 2.12·11-s − 2·13-s + 2.56·17-s + 0.561·19-s + 2.56·21-s + 5.56·23-s + 1.43·27-s + 7.56·29-s + 0.876·31-s − 5.43·33-s + 11.8·37-s − 5.12·39-s − 6.56·41-s + 2.43·43-s + 8.24·47-s + 49-s + 6.56·51-s − 7.12·53-s + 1.43·57-s + 13.3·59-s + 2.87·61-s + 3.56·63-s + 16.1·67-s + 14.2·69-s + ⋯ |
| L(s) = 1 | + 1.47·3-s + 0.377·7-s + 1.18·9-s − 0.640·11-s − 0.554·13-s + 0.621·17-s + 0.128·19-s + 0.558·21-s + 1.15·23-s + 0.276·27-s + 1.40·29-s + 0.157·31-s − 0.946·33-s + 1.94·37-s − 0.820·39-s − 1.02·41-s + 0.371·43-s + 1.20·47-s + 0.142·49-s + 0.918·51-s − 0.978·53-s + 0.190·57-s + 1.74·59-s + 0.368·61-s + 0.448·63-s + 1.96·67-s + 1.71·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.317706169\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.317706169\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 - 2.56T + 3T^{2} \) |
| 11 | \( 1 + 2.12T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 2.56T + 17T^{2} \) |
| 19 | \( 1 - 0.561T + 19T^{2} \) |
| 23 | \( 1 - 5.56T + 23T^{2} \) |
| 29 | \( 1 - 7.56T + 29T^{2} \) |
| 31 | \( 1 - 0.876T + 31T^{2} \) |
| 37 | \( 1 - 11.8T + 37T^{2} \) |
| 41 | \( 1 + 6.56T + 41T^{2} \) |
| 43 | \( 1 - 2.43T + 43T^{2} \) |
| 47 | \( 1 - 8.24T + 47T^{2} \) |
| 53 | \( 1 + 7.12T + 53T^{2} \) |
| 59 | \( 1 - 13.3T + 59T^{2} \) |
| 61 | \( 1 - 2.87T + 61T^{2} \) |
| 67 | \( 1 - 16.1T + 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 + 6.68T + 79T^{2} \) |
| 83 | \( 1 + 13.6T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−8.557796875491366107573358473057, −8.221641239904386592799671074156, −7.48031733045014692840348920895, −6.82955784241169488788817066833, −5.59225864447625533800630998295, −4.78516409494978268553488890310, −3.90411354483070579537160416063, −2.85650336361955474080762333640, −2.46822326680780986771341162871, −1.11911502422421208154117074161,
1.11911502422421208154117074161, 2.46822326680780986771341162871, 2.85650336361955474080762333640, 3.90411354483070579537160416063, 4.78516409494978268553488890310, 5.59225864447625533800630998295, 6.82955784241169488788817066833, 7.48031733045014692840348920895, 8.221641239904386592799671074156, 8.557796875491366107573358473057
