| L(s) = 1 | + 2·3-s + 5-s − 4·7-s + 9-s + 4·13-s + 2·15-s − 4·19-s − 8·21-s + 6·23-s + 25-s − 4·27-s + 6·29-s − 8·31-s − 4·35-s + 2·37-s + 8·39-s − 6·41-s + 8·43-s + 45-s − 6·47-s + 9·49-s − 6·53-s − 8·57-s + 12·59-s − 2·61-s − 4·63-s + 4·65-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s + 1.10·13-s + 0.516·15-s − 0.917·19-s − 1.74·21-s + 1.25·23-s + 1/5·25-s − 0.769·27-s + 1.11·29-s − 1.43·31-s − 0.676·35-s + 0.328·37-s + 1.28·39-s − 0.937·41-s + 1.21·43-s + 0.149·45-s − 0.875·47-s + 9/7·49-s − 0.824·53-s − 1.05·57-s + 1.56·59-s − 0.256·61-s − 0.503·63-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.822643553\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.822643553\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.81773966224960938618699744494, −6.80480566869670209653704136725, −6.52142021339595400974248717701, −5.77668130183311071755397697875, −4.91343059370530248781893008920, −3.71407518242425612975968631850, −3.49316702660849300033786812506, −2.67636874375833584419248148325, −1.98954299847072705724931735624, −0.73611696744318540123035357817,
0.73611696744318540123035357817, 1.98954299847072705724931735624, 2.67636874375833584419248148325, 3.49316702660849300033786812506, 3.71407518242425612975968631850, 4.91343059370530248781893008920, 5.77668130183311071755397697875, 6.52142021339595400974248717701, 6.80480566869670209653704136725, 7.81773966224960938618699744494
