| L(s) = 1 | + 3·3-s − 4·5-s − 4·7-s + 9·9-s + 2·11-s + 13·13-s − 12·15-s − 6·17-s − 36·19-s − 12·21-s + 20·23-s − 109·25-s + 27·27-s + 14·29-s + 152·31-s + 6·33-s + 16·35-s + 258·37-s + 39·39-s + 84·41-s − 188·43-s − 36·45-s − 254·47-s − 327·49-s − 18·51-s − 366·53-s − 8·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.357·5-s − 0.215·7-s + 1/3·9-s + 0.0548·11-s + 0.277·13-s − 0.206·15-s − 0.0856·17-s − 0.434·19-s − 0.124·21-s + 0.181·23-s − 0.871·25-s + 0.192·27-s + 0.0896·29-s + 0.880·31-s + 0.0316·33-s + 0.0772·35-s + 1.14·37-s + 0.160·39-s + 0.319·41-s − 0.666·43-s − 0.119·45-s − 0.788·47-s − 0.953·49-s − 0.0494·51-s − 0.948·53-s − 0.0196·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 13 | \( 1 - p T \) |
| good | 5 | \( 1 + 4 T + p^{3} T^{2} \) |
| 7 | \( 1 + 4 T + p^{3} T^{2} \) |
| 11 | \( 1 - 2 T + p^{3} T^{2} \) |
| 17 | \( 1 + 6 T + p^{3} T^{2} \) |
| 19 | \( 1 + 36 T + p^{3} T^{2} \) |
| 23 | \( 1 - 20 T + p^{3} T^{2} \) |
| 29 | \( 1 - 14 T + p^{3} T^{2} \) |
| 31 | \( 1 - 152 T + p^{3} T^{2} \) |
| 37 | \( 1 - 258 T + p^{3} T^{2} \) |
| 41 | \( 1 - 84 T + p^{3} T^{2} \) |
| 43 | \( 1 + 188 T + p^{3} T^{2} \) |
| 47 | \( 1 + 254 T + p^{3} T^{2} \) |
| 53 | \( 1 + 366 T + p^{3} T^{2} \) |
| 59 | \( 1 - 550 T + p^{3} T^{2} \) |
| 61 | \( 1 - 14 T + p^{3} T^{2} \) |
| 67 | \( 1 - 448 T + p^{3} T^{2} \) |
| 71 | \( 1 + 926 T + p^{3} T^{2} \) |
| 73 | \( 1 - 254 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1328 T + p^{3} T^{2} \) |
| 83 | \( 1 - 186 T + p^{3} T^{2} \) |
| 89 | \( 1 + 336 T + p^{3} T^{2} \) |
| 97 | \( 1 - 614 T + p^{3} T^{2} \) |
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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
−8.199007141859573397779797155115, −7.58885355056681573196717188067, −6.65990914435576323876706701076, −6.01069391225621990630018905738, −4.85877890101840951485711166306, −4.09208728516349750294318093559, −3.27927482062737414944372447471, −2.38445835137681607404900500521, −1.27838234338954444173473271919, 0,
1.27838234338954444173473271919, 2.38445835137681607404900500521, 3.27927482062737414944372447471, 4.09208728516349750294318093559, 4.85877890101840951485711166306, 6.01069391225621990630018905738, 6.65990914435576323876706701076, 7.58885355056681573196717188067, 8.199007141859573397779797155115
