| L(s) = 1 | − 2·2-s + 3-s + 2·4-s − 2·6-s − 2·7-s + 9-s − 3·11-s + 2·12-s + 4·13-s + 4·14-s − 4·16-s + 8·17-s − 2·18-s − 2·21-s + 6·22-s − 23-s − 8·26-s + 27-s − 4·28-s + 29-s − 8·31-s + 8·32-s − 3·33-s − 16·34-s + 2·36-s − 7·37-s + 4·39-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s − 0.816·6-s − 0.755·7-s + 1/3·9-s − 0.904·11-s + 0.577·12-s + 1.10·13-s + 1.06·14-s − 16-s + 1.94·17-s − 0.471·18-s − 0.436·21-s + 1.27·22-s − 0.208·23-s − 1.56·26-s + 0.192·27-s − 0.755·28-s + 0.185·29-s − 1.43·31-s + 1.41·32-s − 0.522·33-s − 2.74·34-s + 1/3·36-s − 1.15·37-s + 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9287179809\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9287179809\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−9.026204258303570972775141705216, −8.393055517263135127517562818868, −7.71321986683429932357831257897, −7.18292635323516030785900459786, −6.12204212796184096175443912425, −5.25685379924244030560862286764, −3.83993387725300169108109456622, −3.09220257677483752161515538695, −1.91602304596019401141580620431, −0.76989224128822694234011319404,
0.76989224128822694234011319404, 1.91602304596019401141580620431, 3.09220257677483752161515538695, 3.83993387725300169108109456622, 5.25685379924244030560862286764, 6.12204212796184096175443912425, 7.18292635323516030785900459786, 7.71321986683429932357831257897, 8.393055517263135127517562818868, 9.026204258303570972775141705216
