| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 5·5-s − 6·6-s + 7·7-s + 8·8-s + 9·9-s − 10·10-s − 11·11-s − 12·12-s − 46·13-s + 14·14-s + 15·15-s + 16·16-s − 126·17-s + 18·18-s + 44·19-s − 20·20-s − 21·21-s − 22·22-s − 96·23-s − 24·24-s + 25·25-s − 92·26-s − 27·27-s + 28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.301·11-s − 0.288·12-s − 0.981·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 1.79·17-s + 0.235·18-s + 0.531·19-s − 0.223·20-s − 0.218·21-s − 0.213·22-s − 0.870·23-s − 0.204·24-s + 1/5·25-s − 0.693·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2310 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.958138096\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.958138096\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + p T \) |
| 7 | \( 1 - p T \) |
| 11 | \( 1 + p T \) |
| good | 13 | \( 1 + 46 T + p^{3} T^{2} \) |
| 17 | \( 1 + 126 T + p^{3} T^{2} \) |
| 19 | \( 1 - 44 T + p^{3} T^{2} \) |
| 23 | \( 1 + 96 T + p^{3} T^{2} \) |
| 29 | \( 1 - 234 T + p^{3} T^{2} \) |
| 31 | \( 1 + 172 T + p^{3} T^{2} \) |
| 37 | \( 1 - 386 T + p^{3} T^{2} \) |
| 41 | \( 1 + 198 T + p^{3} T^{2} \) |
| 43 | \( 1 + 28 T + p^{3} T^{2} \) |
| 47 | \( 1 - 456 T + p^{3} T^{2} \) |
| 53 | \( 1 + 30 T + p^{3} T^{2} \) |
| 59 | \( 1 + 108 T + p^{3} T^{2} \) |
| 61 | \( 1 - 398 T + p^{3} T^{2} \) |
| 67 | \( 1 - 632 T + p^{3} T^{2} \) |
| 71 | \( 1 + 1032 T + p^{3} T^{2} \) |
| 73 | \( 1 + 934 T + p^{3} T^{2} \) |
| 79 | \( 1 - 764 T + p^{3} T^{2} \) |
| 83 | \( 1 - 648 T + p^{3} T^{2} \) |
| 89 | \( 1 + 606 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1550 T + p^{3} T^{2} \) |
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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
−8.520350546598623043791879192613, −7.66761317420609887007382652925, −7.02241500952049819422147304368, −6.26049445128498104896221862132, −5.36953879768541088935745851638, −4.60313165977785000617358342110, −4.12372576372320136972408005971, −2.81777042977448394822591626342, −1.98428640699281477562646957843, −0.55756157323758304789195807812,
0.55756157323758304789195807812, 1.98428640699281477562646957843, 2.81777042977448394822591626342, 4.12372576372320136972408005971, 4.60313165977785000617358342110, 5.36953879768541088935745851638, 6.26049445128498104896221862132, 7.02241500952049819422147304368, 7.66761317420609887007382652925, 8.520350546598623043791879192613
