| L(s) = 1 | + 1.39·2-s − 3-s − 0.0515·4-s − 5-s − 1.39·6-s + 1.30·7-s − 2.86·8-s + 9-s − 1.39·10-s − 0.255·11-s + 0.0515·12-s + 4.21·13-s + 1.82·14-s + 15-s − 3.89·16-s − 3.67·17-s + 1.39·18-s + 1.08·19-s + 0.0515·20-s − 1.30·21-s − 0.356·22-s − 5.23·23-s + 2.86·24-s + 25-s + 5.88·26-s − 27-s − 0.0672·28-s + ⋯ |
| L(s) = 1 | + 0.987·2-s − 0.577·3-s − 0.0257·4-s − 0.447·5-s − 0.569·6-s + 0.493·7-s − 1.01·8-s + 0.333·9-s − 0.441·10-s − 0.0770·11-s + 0.0148·12-s + 1.16·13-s + 0.486·14-s + 0.258·15-s − 0.973·16-s − 0.891·17-s + 0.329·18-s + 0.249·19-s + 0.0115·20-s − 0.284·21-s − 0.0760·22-s − 1.09·23-s + 0.584·24-s + 0.200·25-s + 1.15·26-s − 0.192·27-s − 0.0127·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 - 1.39T + 2T^{2} \) |
| 7 | \( 1 - 1.30T + 7T^{2} \) |
| 11 | \( 1 + 0.255T + 11T^{2} \) |
| 13 | \( 1 - 4.21T + 13T^{2} \) |
| 17 | \( 1 + 3.67T + 17T^{2} \) |
| 19 | \( 1 - 1.08T + 19T^{2} \) |
| 23 | \( 1 + 5.23T + 23T^{2} \) |
| 29 | \( 1 - 2.91T + 29T^{2} \) |
| 31 | \( 1 - 9.05T + 31T^{2} \) |
| 37 | \( 1 + 5.96T + 37T^{2} \) |
| 41 | \( 1 + 2.02T + 41T^{2} \) |
| 43 | \( 1 - 1.49T + 43T^{2} \) |
| 47 | \( 1 - 0.581T + 47T^{2} \) |
| 53 | \( 1 + 8.11T + 53T^{2} \) |
| 59 | \( 1 - 2.64T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 + 1.65T + 67T^{2} \) |
| 71 | \( 1 + 12.1T + 71T^{2} \) |
| 73 | \( 1 + 2.68T + 73T^{2} \) |
| 79 | \( 1 + 0.0164T + 79T^{2} \) |
| 83 | \( 1 - 3.29T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 - 9.96T + 97T^{2} \) |
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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
−7.74363994845184982923751056850, −6.54376243610344088466720019083, −6.34656602762364809740985529359, −5.40875648706250806112501217403, −4.78869831115270726657758716431, −4.16739770849771676265933134330, −3.55552393782270017751154588339, −2.54785711666350263462959075991, −1.29102607031038736854554978517, 0,
1.29102607031038736854554978517, 2.54785711666350263462959075991, 3.55552393782270017751154588339, 4.16739770849771676265933134330, 4.78869831115270726657758716431, 5.40875648706250806112501217403, 6.34656602762364809740985529359, 6.54376243610344088466720019083, 7.74363994845184982923751056850
