| L(s) = 1 | + 10·5-s − 7·7-s − 12·11-s + 30·13-s − 34·17-s − 148·19-s + 152·23-s − 25·25-s + 106·29-s − 304·31-s − 70·35-s − 114·37-s − 202·41-s − 116·43-s + 224·47-s + 49·49-s + 274·53-s − 120·55-s − 660·59-s + 382·61-s + 300·65-s − 12·67-s − 552·71-s − 614·73-s + 84·77-s − 880·79-s − 108·83-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.377·7-s − 0.328·11-s + 0.640·13-s − 0.485·17-s − 1.78·19-s + 1.37·23-s − 1/5·25-s + 0.678·29-s − 1.76·31-s − 0.338·35-s − 0.506·37-s − 0.769·41-s − 0.411·43-s + 0.695·47-s + 1/7·49-s + 0.710·53-s − 0.294·55-s − 1.45·59-s + 0.801·61-s + 0.572·65-s − 0.0218·67-s − 0.922·71-s − 0.984·73-s + 0.124·77-s − 1.25·79-s − 0.142·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + p T \) |
| 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
−8.952956134961024650186597227728, −8.709729886277416171073942789875, −7.35365107341326265859209889398, −6.51007242686440094629244504682, −5.83156664415610907503619128357, −4.85690432321263242348031059647, −3.73652487773864031460674174391, −2.56330860224780271798545205634, −1.58063827633967665063629510653, 0,
1.58063827633967665063629510653, 2.56330860224780271798545205634, 3.73652487773864031460674174391, 4.85690432321263242348031059647, 5.83156664415610907503619128357, 6.51007242686440094629244504682, 7.35365107341326265859209889398, 8.709729886277416171073942789875, 8.952956134961024650186597227728
