| L(s) = 1 | − 3-s + 9-s − 2·11-s − 6·13-s + 4·19-s − 27-s + 2·31-s + 2·33-s + 4·37-s + 6·39-s − 2·41-s − 10·43-s − 6·47-s + 14·53-s − 4·57-s + 4·59-s − 2·61-s + 2·67-s − 6·73-s + 16·79-s + 81-s + 8·83-s − 10·89-s − 2·93-s + 14·97-s − 2·99-s + 101-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.603·11-s − 1.66·13-s + 0.917·19-s − 0.192·27-s + 0.359·31-s + 0.348·33-s + 0.657·37-s + 0.960·39-s − 0.312·41-s − 1.52·43-s − 0.875·47-s + 1.92·53-s − 0.529·57-s + 0.520·59-s − 0.256·61-s + 0.244·67-s − 0.702·73-s + 1.80·79-s + 1/9·81-s + 0.878·83-s − 1.05·89-s − 0.207·93-s + 1.42·97-s − 0.201·99-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−15.35926295422756, −14.90825752822826, −14.54361719646302, −13.65738079937521, −13.36822751429863, −12.70060094853834, −12.11886719090925, −11.77519519440205, −11.27564498658486, −10.46164237429838, −10.07297310628121, −9.659434922096104, −9.016830884686121, −8.155299627007583, −7.736467440607356, −7.091319460271481, −6.673402834080930, −5.852026515419860, −5.122992097554875, −4.992615233414067, −4.152311329196710, −3.294185820489012, −2.607195483515336, −1.921662080419924, −0.8660554877703597, 0,
0.8660554877703597, 1.921662080419924, 2.607195483515336, 3.294185820489012, 4.152311329196710, 4.992615233414067, 5.122992097554875, 5.852026515419860, 6.673402834080930, 7.091319460271481, 7.736467440607356, 8.155299627007583, 9.016830884686121, 9.659434922096104, 10.07297310628121, 10.46164237429838, 11.27564498658486, 11.77519519440205, 12.11886719090925, 12.70060094853834, 13.36822751429863, 13.65738079937521, 14.54361719646302, 14.90825752822826, 15.35926295422756
