| L(s) = 1 | − 5.18·3-s − 5·5-s − 12.0·7-s − 0.0670·9-s − 44.0·11-s − 69.3·13-s + 25.9·15-s + 123.·17-s + 53.0·19-s + 62.3·21-s + 23·23-s + 25·25-s + 140.·27-s + 263.·29-s − 252.·31-s + 228.·33-s + 60.1·35-s + 126.·37-s + 359.·39-s − 65.6·41-s + 120.·43-s + 0.335·45-s + 87.7·47-s − 198.·49-s − 639.·51-s + 640.·53-s + 220.·55-s + ⋯ |
| L(s) = 1 | − 0.998·3-s − 0.447·5-s − 0.649·7-s − 0.00248·9-s − 1.20·11-s − 1.47·13-s + 0.446·15-s + 1.75·17-s + 0.641·19-s + 0.648·21-s + 0.208·23-s + 0.200·25-s + 1.00·27-s + 1.68·29-s − 1.46·31-s + 1.20·33-s + 0.290·35-s + 0.561·37-s + 1.47·39-s − 0.250·41-s + 0.428·43-s + 0.00111·45-s + 0.272·47-s − 0.578·49-s − 1.75·51-s + 1.66·53-s + 0.540·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 + 5.18T + 27T^{2} \) |
| 7 | \( 1 + 12.0T + 343T^{2} \) |
| 11 | \( 1 + 44.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 69.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 123.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 53.0T + 6.85e3T^{2} \) |
| 29 | \( 1 - 263.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 252.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 126.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 65.6T + 6.89e4T^{2} \) |
| 43 | \( 1 - 120.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 87.7T + 1.03e5T^{2} \) |
| 53 | \( 1 - 640.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 109.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 689.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 713.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 819.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.02e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 462.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 197.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.45e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 546.T + 9.12e5T^{2} \) |
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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
−8.368869826429691851356949218169, −7.51704580457486118765838597923, −7.04464147771301268152904080221, −5.84947473415806238492891060820, −5.35834391427856585355721434177, −4.64564422148333643342987009662, −3.28866244953617053060645911961, −2.61242292536146215975202255404, −0.857648842329206787333890148430, 0,
0.857648842329206787333890148430, 2.61242292536146215975202255404, 3.28866244953617053060645911961, 4.64564422148333643342987009662, 5.35834391427856585355721434177, 5.84947473415806238492891060820, 7.04464147771301268152904080221, 7.51704580457486118765838597923, 8.368869826429691851356949218169
