| L(s) = 1 | + 3-s + 5-s − 2·7-s + 9-s + 13-s + 15-s + 6·17-s − 4·19-s − 2·21-s + 25-s + 27-s − 4·29-s + 2·31-s − 2·35-s + 12·37-s + 39-s + 2·41-s − 12·43-s + 45-s + 8·47-s − 3·49-s + 6·51-s + 4·53-s − 4·57-s + 10·61-s − 2·63-s + 65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.277·13-s + 0.258·15-s + 1.45·17-s − 0.917·19-s − 0.436·21-s + 1/5·25-s + 0.192·27-s − 0.742·29-s + 0.359·31-s − 0.338·35-s + 1.97·37-s + 0.160·39-s + 0.312·41-s − 1.82·43-s + 0.149·45-s + 1.16·47-s − 3/7·49-s + 0.840·51-s + 0.549·53-s − 0.529·57-s + 1.28·61-s − 0.251·63-s + 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.550732188\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.550732188\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−13.05233898145106, −12.79834052151204, −12.34284382495927, −11.71522024264784, −11.26494925251752, −10.62467300649634, −10.13372040214285, −9.826710342628782, −9.373480665038852, −8.929578486226697, −8.256490065536789, −7.991737941974066, −7.381764846263736, −6.770063939609452, −6.415516794492977, −5.705420777511296, −5.535766117242067, −4.612500970448180, −4.163549049716120, −3.496286948243229, −3.125218307714208, −2.488021726746587, −1.940768841440331, −1.195274168838887, −0.5402640642349497,
0.5402640642349497, 1.195274168838887, 1.940768841440331, 2.488021726746587, 3.125218307714208, 3.496286948243229, 4.163549049716120, 4.612500970448180, 5.535766117242067, 5.705420777511296, 6.415516794492977, 6.770063939609452, 7.381764846263736, 7.991737941974066, 8.256490065536789, 8.929578486226697, 9.373480665038852, 9.826710342628782, 10.13372040214285, 10.62467300649634, 11.26494925251752, 11.71522024264784, 12.34284382495927, 12.79834052151204, 13.05233898145106
