L(s) = 1 | + (0.955 − 0.294i)2-s + (0.826 − 0.563i)4-s + (−0.988 − 0.149i)5-s + (0.623 + 0.781i)7-s + (0.623 − 0.781i)8-s + (−0.733 + 0.680i)9-s + (−0.988 + 0.149i)10-s + (1.44 + 1.34i)11-s + (−0.367 + 1.61i)13-s + (0.826 + 0.563i)14-s + (0.365 − 0.930i)16-s + (−0.5 + 0.866i)18-s + (−0.365 − 0.632i)19-s + (−0.900 + 0.433i)20-s + (1.78 + 0.858i)22-s + (−0.0747 − 0.997i)23-s + ⋯ |
L(s) = 1 | + (0.955 − 0.294i)2-s + (0.826 − 0.563i)4-s + (−0.988 − 0.149i)5-s + (0.623 + 0.781i)7-s + (0.623 − 0.781i)8-s + (−0.733 + 0.680i)9-s + (−0.988 + 0.149i)10-s + (1.44 + 1.34i)11-s + (−0.367 + 1.61i)13-s + (0.826 + 0.563i)14-s + (0.365 − 0.930i)16-s + (−0.5 + 0.866i)18-s + (−0.365 − 0.632i)19-s + (−0.900 + 0.433i)20-s + (1.78 + 0.858i)22-s + (−0.0747 − 0.997i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.891611475\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.891611475\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.955 + 0.294i)T \) |
| 5 | \( 1 + (0.988 + 0.149i)T \) |
| 7 | \( 1 + (-0.623 - 0.781i)T \) |
good | 3 | \( 1 + (0.733 - 0.680i)T^{2} \) |
| 11 | \( 1 + (-1.44 - 1.34i)T + (0.0747 + 0.997i)T^{2} \) |
| 13 | \( 1 + (0.367 - 1.61i)T + (-0.900 - 0.433i)T^{2} \) |
| 17 | \( 1 + (0.988 + 0.149i)T^{2} \) |
| 19 | \( 1 + (0.365 + 0.632i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.0747 + 0.997i)T + (-0.988 + 0.149i)T^{2} \) |
| 29 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (1.21 + 0.825i)T + (0.365 + 0.930i)T^{2} \) |
| 41 | \( 1 + (-1.19 + 1.49i)T + (-0.222 - 0.974i)T^{2} \) |
| 43 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 47 | \( 1 + (-0.142 + 0.0440i)T + (0.826 - 0.563i)T^{2} \) |
| 53 | \( 1 + (-0.123 + 0.0841i)T + (0.365 - 0.930i)T^{2} \) |
| 59 | \( 1 + (1.23 - 0.185i)T + (0.955 - 0.294i)T^{2} \) |
| 61 | \( 1 + (-0.365 - 0.930i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 73 | \( 1 + (-0.826 - 0.563i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 89 | \( 1 + (-1.32 + 1.22i)T + (0.0747 - 0.997i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.094932979694410414909010794192, −8.904149742476885078774410221189, −7.58007324956040599427168541206, −6.99468957774380041739008543968, −6.20712761374415914960172771845, −5.00443098624922192182420821729, −4.51560000158725957913522153369, −3.87876876064884092499405163717, −2.44248146026443013687532120143, −1.80297873388781097778301920144,
1.12828309492339055015921067880, 3.11643760371794902160959360834, 3.49702789449049218645848291672, 4.27744622895519755223729682263, 5.34979909262003134533549041843, 6.13006394453355250717113882216, 6.84427660324950202397926960702, 7.942535531526313010429715374244, 8.082050633104507371985290664569, 9.099722324783777827444062413984