| L(s) = 1 | + 3·3-s − 3·7-s + 6·9-s + 4·13-s + 7·17-s + 3·19-s − 9·21-s + 9·27-s + 7·29-s − 3·31-s − 9·37-s + 12·39-s + 4·41-s − 6·43-s − 12·47-s + 2·49-s + 21·51-s + 7·53-s + 9·57-s + 12·59-s + 61-s − 18·63-s + 12·67-s − 9·71-s + 2·73-s − 12·79-s + 9·81-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 1.13·7-s + 2·9-s + 1.10·13-s + 1.69·17-s + 0.688·19-s − 1.96·21-s + 1.73·27-s + 1.29·29-s − 0.538·31-s − 1.47·37-s + 1.92·39-s + 0.624·41-s − 0.914·43-s − 1.75·47-s + 2/7·49-s + 2.94·51-s + 0.961·53-s + 1.19·57-s + 1.56·59-s + 0.128·61-s − 2.26·63-s + 1.46·67-s − 1.06·71-s + 0.234·73-s − 1.35·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.092091020\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.092091020\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 7 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−13.19732374566483, −12.68202517439030, −12.47715364047610, −11.69439539971289, −11.32335204475158, −10.37446205087506, −10.04874152716741, −9.846852723594185, −9.311941864685439, −8.657798611440748, −8.461672027822852, −7.995929995972153, −7.406459189842694, −6.833685549778263, −6.626947157084697, −5.675716669050276, −5.432035510156266, −4.529484802468600, −3.828661796687210, −3.520328332964465, −3.053298354247248, −2.800348927894616, −1.819894757277209, −1.367198504532630, −0.6329598523727472,
0.6329598523727472, 1.367198504532630, 1.819894757277209, 2.800348927894616, 3.053298354247248, 3.520328332964465, 3.828661796687210, 4.529484802468600, 5.432035510156266, 5.675716669050276, 6.626947157084697, 6.833685549778263, 7.406459189842694, 7.995929995972153, 8.461672027822852, 8.657798611440748, 9.311941864685439, 9.846852723594185, 10.04874152716741, 10.37446205087506, 11.32335204475158, 11.69439539971289, 12.47715364047610, 12.68202517439030, 13.19732374566483
