| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s + 7-s − 8-s + 9-s − 10-s − 4·11-s − 12-s − 4·13-s − 14-s − 15-s + 16-s + 17-s − 18-s + 20-s − 21-s + 4·22-s + 24-s + 25-s + 4·26-s − 27-s + 28-s − 2·29-s + 30-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 − 1.20·11-s − 0.288·12-s − 1.10·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.223·20-s − 0.218·21-s + 0.852·22-s + 0.204·24-s + 1/5·25-s + 0.784·26-s − 0.192·27-s + 0.188·28-s − 0.371·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.72858694853988, −13.46986736517781, −12.74717521646160, −12.39635445550610, −11.91153690860612, −11.37724669255844, −10.86871952040400, −10.38878070631201, −10.06054082845393, −9.608604860366874, −9.007875303252846, −8.474158593436810, −7.791603873975314, −7.545519184647879, −6.981721344154608, −6.429806365686981, −5.654905659862866, −5.436733386898711, −4.852913108999670, −4.260017335549170, −3.424704366363025, −2.512192335162444, −2.379367798205579, −1.504066664891541, −0.7405553061935173, 0,
0.7405553061935173, 1.504066664891541, 2.379367798205579, 2.512192335162444, 3.424704366363025, 4.260017335549170, 4.852913108999670, 5.436733386898711, 5.654905659862866, 6.429806365686981, 6.981721344154608, 7.545519184647879, 7.791603873975314, 8.474158593436810, 9.007875303252846, 9.608604860366874, 10.06054082845393, 10.38878070631201, 10.86871952040400, 11.37724669255844, 11.91153690860612, 12.39635445550610, 12.74717521646160, 13.46986736517781, 13.72858694853988
