| L(s) = 1 | − 4·3-s − 5·5-s + 16·7-s − 11·9-s − 36·11-s + 42·13-s + 20·15-s − 110·17-s + 116·19-s − 64·21-s + 16·23-s + 25·25-s + 152·27-s − 198·29-s + 240·31-s + 144·33-s − 80·35-s + 258·37-s − 168·39-s + 442·41-s + 292·43-s + 55·45-s + 392·47-s − 87·49-s + 440·51-s − 142·53-s + 180·55-s + ⋯ |
| L(s) = 1 | − 0.769·3-s − 0.447·5-s + 0.863·7-s − 0.407·9-s − 0.986·11-s + 0.896·13-s + 0.344·15-s − 1.56·17-s + 1.40·19-s − 0.665·21-s + 0.145·23-s + 1/5·25-s + 1.08·27-s − 1.26·29-s + 1.39·31-s + 0.759·33-s − 0.386·35-s + 1.14·37-s − 0.689·39-s + 1.68·41-s + 1.03·43-s + 0.182·45-s + 1.21·47-s − 0.253·49-s + 1.20·51-s − 0.368·53-s + 0.441·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.186035278\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.186035278\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + p T \) |
| good | 3 | \( 1 + 4 T + p^{3} T^{2} \) |
| 7 | \( 1 - 16 T + p^{3} T^{2} \) |
| 11 | \( 1 + 36 T + p^{3} T^{2} \) |
| 13 | \( 1 - 42 T + p^{3} T^{2} \) |
| 17 | \( 1 + 110 T + p^{3} T^{2} \) |
| 19 | \( 1 - 116 T + p^{3} T^{2} \) |
| 23 | \( 1 - 16 T + p^{3} T^{2} \) |
| 29 | \( 1 + 198 T + p^{3} T^{2} \) |
| 31 | \( 1 - 240 T + p^{3} T^{2} \) |
| 37 | \( 1 - 258 T + p^{3} T^{2} \) |
| 41 | \( 1 - 442 T + p^{3} T^{2} \) |
| 43 | \( 1 - 292 T + p^{3} T^{2} \) |
| 47 | \( 1 - 392 T + p^{3} T^{2} \) |
| 53 | \( 1 + 142 T + p^{3} T^{2} \) |
| 59 | \( 1 - 348 T + p^{3} T^{2} \) |
| 61 | \( 1 - 570 T + p^{3} T^{2} \) |
| 67 | \( 1 + 692 T + p^{3} T^{2} \) |
| 71 | \( 1 - 168 T + p^{3} T^{2} \) |
| 73 | \( 1 + 134 T + p^{3} T^{2} \) |
| 79 | \( 1 - 784 T + p^{3} T^{2} \) |
| 83 | \( 1 + 564 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1034 T + p^{3} T^{2} \) |
| 97 | \( 1 + 382 T + p^{3} T^{2} \) |
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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
−11.17477835190232058162526978659, −10.74312162761488200144810365859, −9.261958699256468132637762878437, −8.260623451595096834795595279169, −7.44487742774691684407830202821, −6.13261070135051499018771128608, −5.24942133706165827268755822034, −4.24488950968652355784252859585, −2.61448553298652388623298738195, −0.77655187031937956355565516973,
0.77655187031937956355565516973, 2.61448553298652388623298738195, 4.24488950968652355784252859585, 5.24942133706165827268755822034, 6.13261070135051499018771128608, 7.44487742774691684407830202821, 8.260623451595096834795595279169, 9.261958699256468132637762878437, 10.74312162761488200144810365859, 11.17477835190232058162526978659
