| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 5·5-s − 6·6-s − 34·7-s + 8·8-s + 9·9-s − 10·10-s + 28·11-s − 12·12-s − 6·13-s − 68·14-s + 15·15-s + 16·16-s + 8·17-s + 18·18-s + 19·19-s − 20·20-s + 102·21-s + 56·22-s − 204·23-s − 24·24-s + 25·25-s − 12·26-s − 27·27-s − 136·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 1.83·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.767·11-s − 0.288·12-s − 0.128·13-s − 1.29·14-s + 0.258·15-s + 1/4·16-s + 0.114·17-s + 0.235·18-s + 0.229·19-s − 0.223·20-s + 1.05·21-s + 0.542·22-s − 1.84·23-s − 0.204·24-s + 1/5·25-s − 0.0905·26-s − 0.192·27-s − 0.917·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.738571323\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.738571323\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 + p T \) |
| 19 | \( 1 - p T \) |
| good | 7 | \( 1 + 34 T + p^{3} T^{2} \) |
| 11 | \( 1 - 28 T + p^{3} T^{2} \) |
| 13 | \( 1 + 6 T + p^{3} T^{2} \) |
| 17 | \( 1 - 8 T + p^{3} T^{2} \) |
| 23 | \( 1 + 204 T + p^{3} T^{2} \) |
| 29 | \( 1 - 262 T + p^{3} T^{2} \) |
| 31 | \( 1 - 298 T + p^{3} T^{2} \) |
| 37 | \( 1 - 346 T + p^{3} T^{2} \) |
| 41 | \( 1 + 296 T + p^{3} T^{2} \) |
| 43 | \( 1 - 340 T + p^{3} T^{2} \) |
| 47 | \( 1 + 204 T + p^{3} T^{2} \) |
| 53 | \( 1 - 462 T + p^{3} T^{2} \) |
| 59 | \( 1 - 194 T + p^{3} T^{2} \) |
| 61 | \( 1 + 46 T + p^{3} T^{2} \) |
| 67 | \( 1 + 20 T + p^{3} T^{2} \) |
| 71 | \( 1 - 1080 T + p^{3} T^{2} \) |
| 73 | \( 1 + 922 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1382 T + p^{3} T^{2} \) |
| 83 | \( 1 - 10 T + p^{3} T^{2} \) |
| 89 | \( 1 - 180 T + p^{3} T^{2} \) |
| 97 | \( 1 - 514 T + p^{3} T^{2} \) |
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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
−10.11636432727996378614614866025, −9.917596338667617414929765537968, −8.536070279840790171225129170833, −7.30834557457599451453529128593, −6.35796183819578394816221713826, −6.05210708924059362135911429062, −4.54433840481373226682427531291, −3.72833813234566089785756322192, −2.67988902030070968613183996958, −0.72310940816234801702026397946,
0.72310940816234801702026397946, 2.67988902030070968613183996958, 3.72833813234566089785756322192, 4.54433840481373226682427531291, 6.05210708924059362135911429062, 6.35796183819578394816221713826, 7.30834557457599451453529128593, 8.536070279840790171225129170833, 9.917596338667617414929765537968, 10.11636432727996378614614866025
