| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 6·6-s − 24·7-s + 8·8-s + 9·9-s − 12·12-s − 13·13-s − 48·14-s + 16·16-s − 50·17-s + 18·18-s + 28·19-s + 72·21-s + 208·23-s − 24·24-s − 26·26-s − 27·27-s − 96·28-s + 190·29-s + 248·31-s + 32·32-s − 100·34-s + 36·36-s + 186·37-s + 56·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 1.29·7-s + 0.353·8-s + 1/3·9-s − 0.288·12-s − 0.277·13-s − 0.916·14-s + 1/4·16-s − 0.713·17-s + 0.235·18-s + 0.338·19-s + 0.748·21-s + 1.88·23-s − 0.204·24-s − 0.196·26-s − 0.192·27-s − 0.647·28-s + 1.21·29-s + 1.43·31-s + 0.176·32-s − 0.504·34-s + 1/6·36-s + 0.826·37-s + 0.239·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + p T \) |
| good | 7 | \( 1 + 24 T + p^{3} T^{2} \) |
| 11 | \( 1 + p^{3} T^{2} \) |
| 17 | \( 1 + 50 T + p^{3} T^{2} \) |
| 19 | \( 1 - 28 T + p^{3} T^{2} \) |
| 23 | \( 1 - 208 T + p^{3} T^{2} \) |
| 29 | \( 1 - 190 T + p^{3} T^{2} \) |
| 31 | \( 1 - 8 p T + p^{3} T^{2} \) |
| 37 | \( 1 - 186 T + p^{3} T^{2} \) |
| 41 | \( 1 + 194 T + p^{3} T^{2} \) |
| 43 | \( 1 + 348 T + p^{3} T^{2} \) |
| 47 | \( 1 + 260 T + p^{3} T^{2} \) |
| 53 | \( 1 + 462 T + p^{3} T^{2} \) |
| 59 | \( 1 + 520 T + p^{3} T^{2} \) |
| 61 | \( 1 + 506 T + p^{3} T^{2} \) |
| 67 | \( 1 + 772 T + p^{3} T^{2} \) |
| 71 | \( 1 - 780 T + p^{3} T^{2} \) |
| 73 | \( 1 - 62 T + p^{3} T^{2} \) |
| 79 | \( 1 - 736 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1464 T + p^{3} T^{2} \) |
| 89 | \( 1 - 406 T + p^{3} T^{2} \) |
| 97 | \( 1 + 922 T + p^{3} T^{2} \) |
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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
−8.416465990230765079853171416539, −7.31961760683342902126006255534, −6.49625643382066348830568649424, −6.30387285477699337734526995222, −5.03359520233228590046526269434, −4.56863210425791523940489608226, −3.28869187431763498407889215726, −2.75636674314807568986489770775, −1.22818131724662363177845104983, 0,
1.22818131724662363177845104983, 2.75636674314807568986489770775, 3.28869187431763498407889215726, 4.56863210425791523940489608226, 5.03359520233228590046526269434, 6.30387285477699337734526995222, 6.49625643382066348830568649424, 7.31961760683342902126006255534, 8.416465990230765079853171416539
