| L(s) = 1 | − 3·3-s − 5-s − 7-s + 6·9-s − 5·11-s + 5·13-s + 3·15-s − 7·17-s − 2·19-s + 3·21-s + 2·23-s + 25-s − 9·27-s − 7·29-s − 4·31-s + 15·33-s + 35-s + 6·37-s − 15·39-s − 12·41-s − 2·43-s − 6·45-s − 47-s + 49-s + 21·51-s + 5·55-s + 6·57-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.447·5-s − 0.377·7-s + 2·9-s − 1.50·11-s + 1.38·13-s + 0.774·15-s − 1.69·17-s − 0.458·19-s + 0.654·21-s + 0.417·23-s + 1/5·25-s − 1.73·27-s − 1.29·29-s − 0.718·31-s + 2.61·33-s + 0.169·35-s + 0.986·37-s − 2.40·39-s − 1.87·41-s − 0.304·43-s − 0.894·45-s − 0.145·47-s + 1/7·49-s + 2.94·51-s + 0.674·55-s + 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3738745050\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3738745050\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−9.042598540753496698190471958902, −8.205893462202183594605279302674, −7.24732574205984465202036599749, −6.55471779218052889203976082056, −5.91678635331389867682255214840, −5.14013739172314969133240817679, −4.42571840780149440342781486558, −3.43956343442873946903178551051, −1.94747803676559862393917859107, −0.41820752333307382401303841270,
0.41820752333307382401303841270, 1.94747803676559862393917859107, 3.43956343442873946903178551051, 4.42571840780149440342781486558, 5.14013739172314969133240817679, 5.91678635331389867682255214840, 6.55471779218052889203976082056, 7.24732574205984465202036599749, 8.205893462202183594605279302674, 9.042598540753496698190471958902
