| L(s) = 1 | − 3·3-s + 10·5-s + 9·9-s + 12·11-s − 30·13-s − 30·15-s − 34·17-s + 148·19-s − 152·23-s − 25·25-s − 27·27-s − 106·29-s + 304·31-s − 36·33-s − 114·37-s + 90·39-s − 202·41-s − 116·43-s + 90·45-s + 224·47-s + 102·51-s − 274·53-s + 120·55-s − 444·57-s − 660·59-s − 382·61-s − 300·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1/3·9-s + 0.328·11-s − 0.640·13-s − 0.516·15-s − 0.485·17-s + 1.78·19-s − 1.37·23-s − 1/5·25-s − 0.192·27-s − 0.678·29-s + 1.76·31-s − 0.189·33-s − 0.506·37-s + 0.369·39-s − 0.769·41-s − 0.411·43-s + 0.298·45-s + 0.695·47-s + 0.280·51-s − 0.710·53-s + 0.294·55-s − 1.03·57-s − 1.45·59-s − 0.801·61-s − 0.572·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 11 | \( 1 - 12 T + p^{3} T^{2} \) |
| 13 | \( 1 + 30 T + p^{3} T^{2} \) |
| 17 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 19 | \( 1 - 148 T + p^{3} T^{2} \) |
| 23 | \( 1 + 152 T + p^{3} T^{2} \) |
| 29 | \( 1 + 106 T + p^{3} T^{2} \) |
| 31 | \( 1 - 304 T + p^{3} T^{2} \) |
| 37 | \( 1 + 114 T + p^{3} T^{2} \) |
| 41 | \( 1 + 202 T + p^{3} T^{2} \) |
| 43 | \( 1 + 116 T + p^{3} T^{2} \) |
| 47 | \( 1 - 224 T + p^{3} T^{2} \) |
| 53 | \( 1 + 274 T + p^{3} T^{2} \) |
| 59 | \( 1 + 660 T + p^{3} T^{2} \) |
| 61 | \( 1 + 382 T + p^{3} T^{2} \) |
| 67 | \( 1 + 12 T + p^{3} T^{2} \) |
| 71 | \( 1 - 552 T + p^{3} T^{2} \) |
| 73 | \( 1 - 614 T + p^{3} T^{2} \) |
| 79 | \( 1 + 880 T + p^{3} T^{2} \) |
| 83 | \( 1 + 108 T + p^{3} T^{2} \) |
| 89 | \( 1 - 86 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1426 T + p^{3} T^{2} \) |
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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
−8.173446810400435311352064118461, −7.40763674056415617136522378007, −6.55283269048189135007547069779, −5.89696915110052950961235502336, −5.19867559867225855420490867150, −4.38998274964420582607527175776, −3.26033752201566777391972411307, −2.15802732850116304664983305673, −1.27254971812725897729777756503, 0,
1.27254971812725897729777756503, 2.15802732850116304664983305673, 3.26033752201566777391972411307, 4.38998274964420582607527175776, 5.19867559867225855420490867150, 5.89696915110052950961235502336, 6.55283269048189135007547069779, 7.40763674056415617136522378007, 8.173446810400435311352064118461
