| L(s) = 1 | − 2-s + 3·3-s + 4-s + 3·5-s − 3·6-s − 7-s − 8-s + 6·9-s − 3·10-s − 5·11-s + 3·12-s + 13-s + 14-s + 9·15-s + 16-s − 2·17-s − 6·18-s + 4·19-s + 3·20-s − 3·21-s + 5·22-s − 23-s − 3·24-s + 4·25-s − 26-s + 9·27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.73·3-s + 1/2·4-s + 1.34·5-s − 1.22·6-s − 0.377·7-s − 0.353·8-s + 2·9-s − 0.948·10-s − 1.50·11-s + 0.866·12-s + 0.277·13-s + 0.267·14-s + 2.32·15-s + 1/4·16-s − 0.485·17-s − 1.41·18-s + 0.917·19-s + 0.670·20-s − 0.654·21-s + 1.06·22-s − 0.208·23-s − 0.612·24-s + 4/5·25-s − 0.196·26-s + 1.73·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270802 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270802 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.971872800\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.971872800\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−13.01188320218981, −12.52014647668491, −11.98855733539694, −11.05148734938153, −10.74415488741578, −10.27993311435404, −9.864948029710707, −9.360794285144454, −9.184729798716038, −8.798593895887842, −8.032023025111782, −7.837465578801294, −7.341438489116767, −6.869273520292593, −6.222390448634260, −5.602189949607961, −5.364507665935762, −4.489835420418929, −3.858021521039680, −3.274883864594388, −2.654933677803657, −2.435040890043345, −1.981238289550165, −1.323292478369942, −0.5730626878989994,
0.5730626878989994, 1.323292478369942, 1.981238289550165, 2.435040890043345, 2.654933677803657, 3.274883864594388, 3.858021521039680, 4.489835420418929, 5.364507665935762, 5.602189949607961, 6.222390448634260, 6.869273520292593, 7.341438489116767, 7.837465578801294, 8.032023025111782, 8.798593895887842, 9.184729798716038, 9.360794285144454, 9.864948029710707, 10.27993311435404, 10.74415488741578, 11.05148734938153, 11.98855733539694, 12.52014647668491, 13.01188320218981
