| L(s) = 1 | − 3·3-s + 6·5-s + 7·7-s + 9·9-s + 36·11-s + 62·13-s − 18·15-s + 114·17-s − 76·19-s − 21·21-s − 24·23-s − 89·25-s − 27·27-s + 54·29-s − 112·31-s − 108·33-s + 42·35-s − 178·37-s − 186·39-s + 378·41-s − 172·43-s + 54·45-s − 192·47-s + 49·49-s − 342·51-s − 402·53-s + 216·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.536·5-s + 0.377·7-s + 1/3·9-s + 0.986·11-s + 1.32·13-s − 0.309·15-s + 1.62·17-s − 0.917·19-s − 0.218·21-s − 0.217·23-s − 0.711·25-s − 0.192·27-s + 0.345·29-s − 0.648·31-s − 0.569·33-s + 0.202·35-s − 0.790·37-s − 0.763·39-s + 1.43·41-s − 0.609·43-s + 0.178·45-s − 0.595·47-s + 1/7·49-s − 0.939·51-s − 1.04·53-s + 0.529·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.527469426\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.527469426\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 7 | \( 1 - p T \) |
| good | 5 | \( 1 - 6 T + p^{3} T^{2} \) |
| 11 | \( 1 - 36 T + p^{3} T^{2} \) |
| 13 | \( 1 - 62 T + p^{3} T^{2} \) |
| 17 | \( 1 - 114 T + p^{3} T^{2} \) |
| 19 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 23 | \( 1 + 24 T + p^{3} T^{2} \) |
| 29 | \( 1 - 54 T + p^{3} T^{2} \) |
| 31 | \( 1 + 112 T + p^{3} T^{2} \) |
| 37 | \( 1 + 178 T + p^{3} T^{2} \) |
| 41 | \( 1 - 378 T + p^{3} T^{2} \) |
| 43 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 47 | \( 1 + 192 T + p^{3} T^{2} \) |
| 53 | \( 1 + 402 T + p^{3} T^{2} \) |
| 59 | \( 1 - 396 T + p^{3} T^{2} \) |
| 61 | \( 1 - 254 T + p^{3} T^{2} \) |
| 67 | \( 1 + 1012 T + p^{3} T^{2} \) |
| 71 | \( 1 - 840 T + p^{3} T^{2} \) |
| 73 | \( 1 - 890 T + p^{3} T^{2} \) |
| 79 | \( 1 - 80 T + p^{3} T^{2} \) |
| 83 | \( 1 + 108 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1638 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1010 T + p^{3} T^{2} \) |
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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
−13.83201726133467512318490119763, −12.61856591791949487538252939790, −11.58908163730096665968715601934, −10.58315257361744133740154201358, −9.426228672939652439237526663022, −8.114389789415304142426981918414, −6.51779561773031791860008209629, −5.56030199737424878272255152376, −3.86728510554865626200271519258, −1.42982332922689640035017855530,
1.42982332922689640035017855530, 3.86728510554865626200271519258, 5.56030199737424878272255152376, 6.51779561773031791860008209629, 8.114389789415304142426981918414, 9.426228672939652439237526663022, 10.58315257361744133740154201358, 11.58908163730096665968715601934, 12.61856591791949487538252939790, 13.83201726133467512318490119763
