| L(s) = 1 | − 4.84·2-s + 3·3-s + 15.5·4-s + 5·5-s − 14.5·6-s − 26.0·7-s − 36.4·8-s + 9·9-s − 24.2·10-s + 46.5·12-s − 14.0·13-s + 126.·14-s + 15·15-s + 52.6·16-s + 42.1·17-s − 43.6·18-s − 107.·19-s + 77.5·20-s − 78.2·21-s + 16.3·23-s − 109.·24-s + 25·25-s + 68.3·26-s + 27·27-s − 404.·28-s + 153.·29-s − 72.7·30-s + ⋯ |
| L(s) = 1 | − 1.71·2-s + 0.577·3-s + 1.93·4-s + 0.447·5-s − 0.989·6-s − 1.40·7-s − 1.61·8-s + 0.333·9-s − 0.766·10-s + 1.12·12-s − 0.300·13-s + 2.41·14-s + 0.258·15-s + 0.823·16-s + 0.602·17-s − 0.571·18-s − 1.30·19-s + 0.867·20-s − 0.813·21-s + 0.148·23-s − 0.930·24-s + 0.200·25-s + 0.515·26-s + 0.192·27-s − 2.73·28-s + 0.984·29-s − 0.442·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - 3T \) |
| 5 | \( 1 - 5T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 4.84T + 8T^{2} \) |
| 7 | \( 1 + 26.0T + 343T^{2} \) |
| 13 | \( 1 + 14.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 42.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 107.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 16.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 153.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 12.3T + 2.97e4T^{2} \) |
| 37 | \( 1 - 82.6T + 5.06e4T^{2} \) |
| 41 | \( 1 + 309.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 371.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 286.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 353.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 106.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 884.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 856.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 36.4T + 3.57e5T^{2} \) |
| 73 | \( 1 - 974.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.03e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 552.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.18e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.20e3T + 9.12e5T^{2} \) |
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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
−8.537749218395779361234620976361, −8.093848740001972078324811771562, −6.87766258638717576921493567247, −6.71254354736699497160679393543, −5.62820203498933254899558485076, −4.10261193558382492081923306073, −2.90859897830017552979620570010, −2.28939218965778785060848951432, −1.07319220290294807743428537787, 0,
1.07319220290294807743428537787, 2.28939218965778785060848951432, 2.90859897830017552979620570010, 4.10261193558382492081923306073, 5.62820203498933254899558485076, 6.71254354736699497160679393543, 6.87766258638717576921493567247, 8.093848740001972078324811771562, 8.537749218395779361234620976361
