L(s) = 1 | + 2·3-s + 4·5-s − 7-s + 9-s + 8·15-s − 2·17-s − 2·19-s − 2·21-s − 8·23-s + 11·25-s − 4·27-s − 2·29-s − 4·31-s − 4·35-s + 6·37-s − 2·41-s + 8·43-s + 4·45-s + 4·47-s + 49-s − 4·51-s + 10·53-s − 4·57-s + 6·59-s − 4·61-s − 63-s − 12·67-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 1.78·5-s − 0.377·7-s + 1/3·9-s + 2.06·15-s − 0.485·17-s − 0.458·19-s − 0.436·21-s − 1.66·23-s + 11/5·25-s − 0.769·27-s − 0.371·29-s − 0.718·31-s − 0.676·35-s + 0.986·37-s − 0.312·41-s + 1.21·43-s + 0.596·45-s + 0.583·47-s + 1/7·49-s − 0.560·51-s + 1.37·53-s − 0.529·57-s + 0.781·59-s − 0.512·61-s − 0.125·63-s − 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.385221218\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.385221218\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−10.76672173106036800914090760816, −9.957478933829252598803944087984, −9.271787274682453081680950548186, −8.676248321093563858087492291911, −7.51048678322729423104589438688, −6.28154185525062265042607550144, −5.61936194790385614498397188009, −4.05975063928059999821924523382, −2.66547510967810892791353271494, −1.94068717855646026245068222948,
1.94068717855646026245068222948, 2.66547510967810892791353271494, 4.05975063928059999821924523382, 5.61936194790385614498397188009, 6.28154185525062265042607550144, 7.51048678322729423104589438688, 8.676248321093563858087492291911, 9.271787274682453081680950548186, 9.957478933829252598803944087984, 10.76672173106036800914090760816