| L(s) = 1 | + 3-s + 3.93·5-s + 7-s + 9-s − 2.69·11-s − 1.59·13-s + 3.93·15-s − 1.68·17-s − 2.69·19-s + 21-s + 23-s + 10.4·25-s + 27-s + 4.39·29-s + 8.29·31-s − 2.69·33-s + 3.93·35-s + 6.24·37-s − 1.59·39-s + 2.60·41-s − 10.2·43-s + 3.93·45-s + 3.07·47-s + 49-s − 1.68·51-s + 2.38·53-s − 10.6·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.75·5-s + 0.377·7-s + 0.333·9-s − 0.813·11-s − 0.441·13-s + 1.01·15-s − 0.407·17-s − 0.619·19-s + 0.218·21-s + 0.208·23-s + 2.09·25-s + 0.192·27-s + 0.816·29-s + 1.49·31-s − 0.469·33-s + 0.664·35-s + 1.02·37-s − 0.254·39-s + 0.406·41-s − 1.56·43-s + 0.586·45-s + 0.448·47-s + 0.142·49-s − 0.235·51-s + 0.328·53-s − 1.43·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.813270954\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.813270954\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - 3.93T + 5T^{2} \) |
| 11 | \( 1 + 2.69T + 11T^{2} \) |
| 13 | \( 1 + 1.59T + 13T^{2} \) |
| 17 | \( 1 + 1.68T + 17T^{2} \) |
| 19 | \( 1 + 2.69T + 19T^{2} \) |
| 29 | \( 1 - 4.39T + 29T^{2} \) |
| 31 | \( 1 - 8.29T + 31T^{2} \) |
| 37 | \( 1 - 6.24T + 37T^{2} \) |
| 41 | \( 1 - 2.60T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 - 3.07T + 47T^{2} \) |
| 53 | \( 1 - 2.38T + 53T^{2} \) |
| 59 | \( 1 + 1.63T + 59T^{2} \) |
| 61 | \( 1 + 1.29T + 61T^{2} \) |
| 67 | \( 1 + 11.0T + 67T^{2} \) |
| 71 | \( 1 - 7.70T + 71T^{2} \) |
| 73 | \( 1 - 9.36T + 73T^{2} \) |
| 79 | \( 1 - 7.54T + 79T^{2} \) |
| 83 | \( 1 + 2.16T + 83T^{2} \) |
| 89 | \( 1 - 5.84T + 89T^{2} \) |
| 97 | \( 1 - 7.64T + 97T^{2} \) |
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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
−7.990300906422095995097169176002, −7.13681562379016719633513654462, −6.36282130527423968405966198847, −5.90982071378046167217577919912, −4.85045896170326576720347646699, −4.68053138656830208834788849609, −3.25504298334978458034956427917, −2.42055117258442146790351930221, −2.09487490023616780842556930740, −0.966685214681881007894364630013,
0.966685214681881007894364630013, 2.09487490023616780842556930740, 2.42055117258442146790351930221, 3.25504298334978458034956427917, 4.68053138656830208834788849609, 4.85045896170326576720347646699, 5.90982071378046167217577919912, 6.36282130527423968405966198847, 7.13681562379016719633513654462, 7.990300906422095995097169176002
