| L(s) = 1 | − 3-s − 4.14·5-s − 3.38·7-s + 9-s + 5.90·11-s − 3.90·13-s + 4.14·15-s − 3.90·17-s + 0.388·19-s + 3.38·21-s − 0.147·23-s + 12.2·25-s − 27-s + 8·29-s − 2.51·31-s − 5.90·33-s + 14.0·35-s + 4.61·37-s + 3.90·39-s + 12.0·41-s − 3.77·43-s − 4.14·45-s + 8.68·47-s + 4.48·49-s + 3.90·51-s + 5.27·53-s − 24.4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.85·5-s − 1.28·7-s + 0.333·9-s + 1.78·11-s − 1.08·13-s + 1.07·15-s − 0.947·17-s + 0.0891·19-s + 0.739·21-s − 0.0307·23-s + 2.44·25-s − 0.192·27-s + 1.48·29-s − 0.452·31-s − 1.02·33-s + 2.37·35-s + 0.758·37-s + 0.625·39-s + 1.88·41-s − 0.576·43-s − 0.618·45-s + 1.26·47-s + 0.640·49-s + 0.546·51-s + 0.724·53-s − 3.30·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6143552972\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6143552972\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 67 | \( 1 + T \) |
| good | 5 | \( 1 + 4.14T + 5T^{2} \) |
| 7 | \( 1 + 3.38T + 7T^{2} \) |
| 11 | \( 1 - 5.90T + 11T^{2} \) |
| 13 | \( 1 + 3.90T + 13T^{2} \) |
| 17 | \( 1 + 3.90T + 17T^{2} \) |
| 19 | \( 1 - 0.388T + 19T^{2} \) |
| 23 | \( 1 + 0.147T + 23T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 + 2.51T + 31T^{2} \) |
| 37 | \( 1 - 4.61T + 37T^{2} \) |
| 41 | \( 1 - 12.0T + 41T^{2} \) |
| 43 | \( 1 + 3.77T + 43T^{2} \) |
| 47 | \( 1 - 8.68T + 47T^{2} \) |
| 53 | \( 1 - 5.27T + 53T^{2} \) |
| 59 | \( 1 + 1.27T + 59T^{2} \) |
| 61 | \( 1 - 7.42T + 61T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 + 16.8T + 73T^{2} \) |
| 79 | \( 1 + 9.51T + 79T^{2} \) |
| 83 | \( 1 - 0.723T + 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 - 6.38T + 97T^{2} \) |
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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
−10.33931853572367133136981587473, −9.346132137489192261147862407517, −8.667630078394018464895348068929, −7.38647210100052798063885552573, −6.91631429524949064596600362752, −6.10085488248652617837407547233, −4.48510662173206717410088212176, −4.05352112105265193005564674224, −2.92794107556898689484068298267, −0.64217678266333529459716090872,
0.64217678266333529459716090872, 2.92794107556898689484068298267, 4.05352112105265193005564674224, 4.48510662173206717410088212176, 6.10085488248652617837407547233, 6.91631429524949064596600362752, 7.38647210100052798063885552573, 8.667630078394018464895348068929, 9.346132137489192261147862407517, 10.33931853572367133136981587473
