| L(s) = 1 | − 3-s − 2·7-s + 9-s − 5·11-s + 5·13-s − 17-s + 2·19-s + 2·21-s + 23-s − 27-s + 8·29-s + 7·31-s + 5·33-s + 4·37-s − 5·39-s + 9·41-s + 43-s + 8·47-s − 3·49-s + 51-s + 5·53-s − 2·57-s + 4·59-s + 4·61-s − 2·63-s + 11·67-s − 69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s − 1.50·11-s + 1.38·13-s − 0.242·17-s + 0.458·19-s + 0.436·21-s + 0.208·23-s − 0.192·27-s + 1.48·29-s + 1.25·31-s + 0.870·33-s + 0.657·37-s − 0.800·39-s + 1.40·41-s + 0.152·43-s + 1.16·47-s − 3/7·49-s + 0.140·51-s + 0.686·53-s − 0.264·57-s + 0.520·59-s + 0.512·61-s − 0.251·63-s + 1.34·67-s − 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.014319053\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.014319053\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−14.33733859346421, −13.85538159318920, −13.38717001867149, −12.93080584043308, −12.56151410211134, −11.95304674748878, −11.25126031858968, −10.95968306821438, −10.34609981744473, −9.967556348057843, −9.414702234077682, −8.630776787159064, −8.234727192008934, −7.651648632328431, −6.981664727833977, −6.360952960447146, −6.035745492367891, −5.370837903678800, −4.840679449565464, −4.135288184223970, −3.500796172544208, −2.736194984213660, −2.319635753204605, −0.9989952995424458, −0.6497729768301859,
0.6497729768301859, 0.9989952995424458, 2.319635753204605, 2.736194984213660, 3.500796172544208, 4.135288184223970, 4.840679449565464, 5.370837903678800, 6.035745492367891, 6.360952960447146, 6.981664727833977, 7.651648632328431, 8.234727192008934, 8.630776787159064, 9.414702234077682, 9.967556348057843, 10.34609981744473, 10.95968306821438, 11.25126031858968, 11.95304674748878, 12.56151410211134, 12.93080584043308, 13.38717001867149, 13.85538159318920, 14.33733859346421
