L(s) = 1 | + 2-s + 4-s − 5-s + 2·7-s + 8-s − 10-s + 4·11-s − 4·13-s + 2·14-s + 16-s + 6·17-s − 20-s + 4·22-s + 25-s − 4·26-s + 2·28-s − 4·29-s + 31-s + 32-s + 6·34-s − 2·35-s + 4·37-s − 40-s + 6·41-s − 8·43-s + 4·44-s + 12·47-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.755·7-s + 0.353·8-s − 0.316·10-s + 1.20·11-s − 1.10·13-s + 0.534·14-s + 1/4·16-s + 1.45·17-s − 0.223·20-s + 0.852·22-s + 1/5·25-s − 0.784·26-s + 0.377·28-s − 0.742·29-s + 0.179·31-s + 0.176·32-s + 1.02·34-s − 0.338·35-s + 0.657·37-s − 0.158·40-s + 0.937·41-s − 1.21·43-s + 0.603·44-s + 1.75·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.185478637\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.185478637\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 - T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.747416751953644846923547814896, −7.74335668534770922509830429662, −7.42070250982822790241988821325, −6.46093259456322875870891199741, −5.58739815873380500661416063334, −4.84972388474025854571339429019, −4.08822080273047061072314378148, −3.32306644259935075748578991415, −2.19359530048026004945282081289, −1.06631071912869019020178508767,
1.06631071912869019020178508767, 2.19359530048026004945282081289, 3.32306644259935075748578991415, 4.08822080273047061072314378148, 4.84972388474025854571339429019, 5.58739815873380500661416063334, 6.46093259456322875870891199741, 7.42070250982822790241988821325, 7.74335668534770922509830429662, 8.747416751953644846923547814896