| L(s) = 1 | + 3-s + 9-s − 2·11-s + 3·13-s − 6·17-s − 7·19-s − 6·23-s + 27-s − 2·29-s − 5·31-s − 2·33-s + 10·37-s + 3·39-s − 12·41-s − 3·43-s − 10·47-s − 6·51-s − 7·57-s − 6·59-s + 13·61-s − 7·67-s − 6·69-s + 4·71-s + 6·73-s + 8·79-s + 81-s − 6·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 0.603·11-s + 0.832·13-s − 1.45·17-s − 1.60·19-s − 1.25·23-s + 0.192·27-s − 0.371·29-s − 0.898·31-s − 0.348·33-s + 1.64·37-s + 0.480·39-s − 1.87·41-s − 0.457·43-s − 1.45·47-s − 0.840·51-s − 0.927·57-s − 0.781·59-s + 1.66·61-s − 0.855·67-s − 0.722·69-s + 0.474·71-s + 0.702·73-s + 0.900·79-s + 1/9·81-s − 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.152028012\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.152028012\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−14.33987259213176, −13.69576192046572, −13.37997869624243, −12.81062074771460, −12.65391781237376, −11.64989186684273, −11.22462153649857, −10.84726085586320, −10.13228101583562, −9.820087142327520, −9.030599442443608, −8.567967517505266, −8.243224278242111, −7.711198347152209, −6.926417156310173, −6.424878467553990, −6.044567425130844, −5.190118668014519, −4.565791876307261, −4.010439874358238, −3.531598865748322, −2.683654113654562, −2.042318483691804, −1.657442737851023, −0.3278281957453627,
0.3278281957453627, 1.657442737851023, 2.042318483691804, 2.683654113654562, 3.531598865748322, 4.010439874358238, 4.565791876307261, 5.190118668014519, 6.044567425130844, 6.424878467553990, 6.926417156310173, 7.711198347152209, 8.243224278242111, 8.567967517505266, 9.030599442443608, 9.820087142327520, 10.13228101583562, 10.84726085586320, 11.22462153649857, 11.64989186684273, 12.65391781237376, 12.81062074771460, 13.37997869624243, 13.69576192046572, 14.33987259213176
