L(s) = 1 | + 2·3-s + 2·5-s − 7-s + 9-s + 6·11-s + 4·13-s + 4·15-s − 2·17-s + 4·19-s − 2·21-s − 23-s − 25-s − 4·27-s + 10·29-s + 8·31-s + 12·33-s − 2·35-s + 8·37-s + 8·39-s − 2·41-s + 6·43-s + 2·45-s − 12·47-s + 49-s − 4·51-s − 12·53-s + 12·55-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s + 1.80·11-s + 1.10·13-s + 1.03·15-s − 0.485·17-s + 0.917·19-s − 0.436·21-s − 0.208·23-s − 1/5·25-s − 0.769·27-s + 1.85·29-s + 1.43·31-s + 2.08·33-s − 0.338·35-s + 1.31·37-s + 1.28·39-s − 0.312·41-s + 0.914·43-s + 0.298·45-s − 1.75·47-s + 1/7·49-s − 0.560·51-s − 1.64·53-s + 1.61·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.795288340\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.795288340\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−16.53132312073865, −15.92153013273109, −15.49252446291654, −14.57615126390428, −14.29117224664015, −13.72434432318716, −13.48457644554681, −12.76563062661551, −11.80992574341641, −11.57103662430554, −10.64537252641621, −9.732646485817622, −9.582604694984125, −8.944738868743051, −8.394658352436116, −7.839077177662720, −6.772658302921117, −6.294478245188025, −5.922402974633570, −4.676589585037085, −4.045354451808486, −3.230122360089137, −2.738243417114481, −1.691145904993036, −1.079703893022837,
1.079703893022837, 1.691145904993036, 2.738243417114481, 3.230122360089137, 4.045354451808486, 4.676589585037085, 5.922402974633570, 6.294478245188025, 6.772658302921117, 7.839077177662720, 8.394658352436116, 8.944738868743051, 9.582604694984125, 9.732646485817622, 10.64537252641621, 11.57103662430554, 11.80992574341641, 12.76563062661551, 13.48457644554681, 13.72434432318716, 14.29117224664015, 14.57615126390428, 15.49252446291654, 15.92153013273109, 16.53132312073865