| L(s) = 1 | + 1.41·2-s + 5.41·3-s + 2.00·4-s + 7.66·6-s + 8.24i·7-s + 2.82·8-s + 20.3·9-s − 15.8i·11-s + 10.8·12-s + 14.3·13-s + 11.6i·14-s + 4.00·16-s − 10.1i·17-s + 28.8·18-s + 36.5i·19-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.80·3-s + 0.500·4-s + 1.27·6-s + 1.17i·7-s + 0.353·8-s + 2.26·9-s − 1.43i·11-s + 0.903·12-s + 1.10·13-s + 0.832i·14-s + 0.250·16-s − 0.598i·17-s + 1.60·18-s + 1.92i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.254i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.967 - 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(6.190633321\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.190633321\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41T \) |
| 5 | \( 1 \) |
| 23 | \( 1 + (22.2 - 5.84i)T \) |
| good | 3 | \( 1 - 5.41T + 9T^{2} \) |
| 7 | \( 1 - 8.24iT - 49T^{2} \) |
| 11 | \( 1 + 15.8iT - 121T^{2} \) |
| 13 | \( 1 - 14.3T + 169T^{2} \) |
| 17 | \( 1 + 10.1iT - 289T^{2} \) |
| 19 | \( 1 - 36.5iT - 361T^{2} \) |
| 29 | \( 1 - 6.46T + 841T^{2} \) |
| 31 | \( 1 + 42.8T + 961T^{2} \) |
| 37 | \( 1 + 63.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 37.0T + 1.68e3T^{2} \) |
| 43 | \( 1 - 6.00iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 32.4T + 2.20e3T^{2} \) |
| 53 | \( 1 + 36.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 6.65T + 3.48e3T^{2} \) |
| 61 | \( 1 - 55.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 4.45iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 118.T + 5.04e3T^{2} \) |
| 73 | \( 1 + 82.2T + 5.32e3T^{2} \) |
| 79 | \( 1 - 133. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 67.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 104. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 98.6iT - 9.40e3T^{2} \) |
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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
−9.387093883104077784205286350868, −8.621466427148635366243635176159, −8.251661581820115716890254808268, −7.35166668896175988330536979719, −6.04738427542628774486779931234, −5.53444231136680581405212364948, −3.83179674627222010692931487921, −3.55239191871729197265280203428, −2.51340678376720270134607423694, −1.61616000945144785645389214898,
1.42859721503050890380026538747, 2.37400186329977045100826373228, 3.48027429779350256456649191406, 4.11523090886801972962849658472, 4.82301240113152494202476449705, 6.56945306468852791396051548547, 7.14022009458818865604263375062, 7.85787256388513203964993583374, 8.679980733576925226733728038072, 9.541834561247914735249234486539
