| L(s) = 1 | − 0.0987·3-s + 0.748·5-s + 1.53·7-s − 2.99·9-s + 11-s + 0.342·13-s − 0.0739·15-s − 0.815·17-s + 2.91·19-s − 0.151·21-s + 4.33·23-s − 4.43·25-s + 0.591·27-s − 7.03·29-s − 10.4·31-s − 0.0987·33-s + 1.15·35-s − 2.05·37-s − 0.0338·39-s + 2.01·41-s − 10.4·43-s − 2.23·45-s + 3.96·47-s − 4.64·49-s + 0.0804·51-s − 2.91·53-s + 0.748·55-s + ⋯ |
| L(s) = 1 | − 0.0569·3-s + 0.334·5-s + 0.580·7-s − 0.996·9-s + 0.301·11-s + 0.0951·13-s − 0.0190·15-s − 0.197·17-s + 0.669·19-s − 0.0330·21-s + 0.902·23-s − 0.887·25-s + 0.113·27-s − 1.30·29-s − 1.86·31-s − 0.0171·33-s + 0.194·35-s − 0.337·37-s − 0.00542·39-s + 0.315·41-s − 1.59·43-s − 0.333·45-s + 0.577·47-s − 0.662·49-s + 0.0112·51-s − 0.399·53-s + 0.100·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 137 | \( 1 + T \) |
| good | 3 | \( 1 + 0.0987T + 3T^{2} \) |
| 5 | \( 1 - 0.748T + 5T^{2} \) |
| 7 | \( 1 - 1.53T + 7T^{2} \) |
| 13 | \( 1 - 0.342T + 13T^{2} \) |
| 17 | \( 1 + 0.815T + 17T^{2} \) |
| 19 | \( 1 - 2.91T + 19T^{2} \) |
| 23 | \( 1 - 4.33T + 23T^{2} \) |
| 29 | \( 1 + 7.03T + 29T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 + 2.05T + 37T^{2} \) |
| 41 | \( 1 - 2.01T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 - 3.96T + 47T^{2} \) |
| 53 | \( 1 + 2.91T + 53T^{2} \) |
| 59 | \( 1 - 9.20T + 59T^{2} \) |
| 61 | \( 1 - 9.38T + 61T^{2} \) |
| 67 | \( 1 + 6.51T + 67T^{2} \) |
| 71 | \( 1 + 3.77T + 71T^{2} \) |
| 73 | \( 1 + 13.7T + 73T^{2} \) |
| 79 | \( 1 - 7.39T + 79T^{2} \) |
| 83 | \( 1 - 4.28T + 83T^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 + 9.99T + 97T^{2} \) |
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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
−7.70027335393136355985333834387, −7.08899518910002570043055686786, −6.21591013242634861305589057370, −5.43196102443004641762091043490, −5.13325055920897360002672945237, −3.93439545274891387551869931972, −3.27879453996087923788686581466, −2.23227611151014377868422980142, −1.43617656333899318230465460059, 0,
1.43617656333899318230465460059, 2.23227611151014377868422980142, 3.27879453996087923788686581466, 3.93439545274891387551869931972, 5.13325055920897360002672945237, 5.43196102443004641762091043490, 6.21591013242634861305589057370, 7.08899518910002570043055686786, 7.70027335393136355985333834387
