| L(s) = 1 | + 8·4-s − 37i·7-s + 70i·13-s + 64·16-s + 163·19-s − 296i·28-s − 19·31-s − 433i·37-s − 449i·43-s − 1.02e3·49-s + 560i·52-s + 719·61-s + 512·64-s − 880i·67-s + 271i·73-s + ⋯ |
| L(s) = 1 | + 4-s − 1.99i·7-s + 1.49i·13-s + 16-s + 1.96·19-s − 1.99i·28-s − 0.110·31-s − 1.92i·37-s − 1.59i·43-s − 2.99·49-s + 1.49i·52-s + 1.50·61-s + 64-s − 1.60i·67-s + 0.434i·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.701008151\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.701008151\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 8T^{2} \) |
| 7 | \( 1 + 37iT - 343T^{2} \) |
| 11 | \( 1 + 1.33e3T^{2} \) |
| 13 | \( 1 - 70iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 4.91e3T^{2} \) |
| 19 | \( 1 - 163T + 6.85e3T^{2} \) |
| 23 | \( 1 - 1.21e4T^{2} \) |
| 29 | \( 1 + 2.43e4T^{2} \) |
| 31 | \( 1 + 19T + 2.97e4T^{2} \) |
| 37 | \( 1 + 433iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 6.89e4T^{2} \) |
| 43 | \( 1 + 449iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 1.03e5T^{2} \) |
| 53 | \( 1 - 1.48e5T^{2} \) |
| 59 | \( 1 + 2.05e5T^{2} \) |
| 61 | \( 1 - 719T + 2.26e5T^{2} \) |
| 67 | \( 1 + 880iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 3.57e5T^{2} \) |
| 73 | \( 1 - 271iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 503T + 4.93e5T^{2} \) |
| 83 | \( 1 - 5.71e5T^{2} \) |
| 89 | \( 1 + 7.04e5T^{2} \) |
| 97 | \( 1 + 523iT - 9.12e5T^{2} \) |
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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
−10.07632922561253161996693268573, −9.289817213313369317696058893857, −7.84586098052872572990272910799, −7.15096164573269241705698939406, −6.79041699374872039093559715349, −5.46370834143739752590778046173, −4.16731921881620373603199531963, −3.41007351263224283719934570235, −1.87554446615192534822022068435, −0.800101163494392080922744910495,
1.31892464296328645811167867345, 2.71628928804755121360437384943, 3.11546848500810103180832529627, 5.22771646484566273395918750131, 5.65050733363317889129087755091, 6.58555901472753891567990605024, 7.77881913353151303155891928591, 8.355118521535770959554939520664, 9.494452201942319938462188342780, 10.17086911743834983054082781157
