| L(s) = 1 | − 2.35·3-s + 2.56·5-s − 3.68·7-s + 2.56·9-s − 1.03·11-s − 6.04·15-s + 6.12·17-s + 5.00·19-s + 8.68·21-s − 7.07·23-s + 1.56·25-s + 1.03·27-s − 5·29-s + 3.39·31-s + 2.43·33-s − 9.43·35-s − 2.12·37-s + 4.12·41-s + 8.39·43-s + 6.56·45-s − 10.7·47-s + 6.56·49-s − 14.4·51-s − 2.56·53-s − 2.64·55-s − 11.8·57-s − 10.4·59-s + ⋯ |
| L(s) = 1 | − 1.36·3-s + 1.14·5-s − 1.39·7-s + 0.853·9-s − 0.311·11-s − 1.55·15-s + 1.48·17-s + 1.14·19-s + 1.89·21-s − 1.47·23-s + 0.312·25-s + 0.198·27-s − 0.928·29-s + 0.609·31-s + 0.424·33-s − 1.59·35-s − 0.349·37-s + 0.643·41-s + 1.28·43-s + 0.978·45-s − 1.56·47-s + 0.937·49-s − 2.02·51-s − 0.351·53-s − 0.357·55-s − 1.56·57-s − 1.36·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9884544998\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9884544998\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 2.35T + 3T^{2} \) |
| 5 | \( 1 - 2.56T + 5T^{2} \) |
| 7 | \( 1 + 3.68T + 7T^{2} \) |
| 11 | \( 1 + 1.03T + 11T^{2} \) |
| 17 | \( 1 - 6.12T + 17T^{2} \) |
| 19 | \( 1 - 5.00T + 19T^{2} \) |
| 23 | \( 1 + 7.07T + 23T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 - 3.39T + 31T^{2} \) |
| 37 | \( 1 + 2.12T + 37T^{2} \) |
| 41 | \( 1 - 4.12T + 41T^{2} \) |
| 43 | \( 1 - 8.39T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 + 2.56T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 - 9.72T + 67T^{2} \) |
| 71 | \( 1 + 5.00T + 71T^{2} \) |
| 73 | \( 1 - 4.31T + 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 - 2.64T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 - 9.80T + 97T^{2} \) |
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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
−7.966978793248329150416043281509, −7.31648007063594530399840651023, −6.32584151595420239898285904954, −6.01252623611354371405459961276, −5.57912235069714940301745037323, −4.82574071743881579282654378933, −3.61965786381508758424748331480, −2.88533637917719291020499998240, −1.69071349775762249863402761818, −0.57456093185557302966688042000,
0.57456093185557302966688042000, 1.69071349775762249863402761818, 2.88533637917719291020499998240, 3.61965786381508758424748331480, 4.82574071743881579282654378933, 5.57912235069714940301745037323, 6.01252623611354371405459961276, 6.32584151595420239898285904954, 7.31648007063594530399840651023, 7.966978793248329150416043281509
