L(s) = 1 | − 2·4-s + 7-s − 4·13-s + 4·16-s − 3·17-s + 19-s − 3·23-s − 2·28-s − 9·29-s − 10·31-s + 4·37-s + 6·41-s − 43-s − 6·47-s + 49-s + 8·52-s − 9·53-s − 15·59-s + 13·61-s − 8·64-s − 2·67-s + 6·68-s + 6·71-s − 16·73-s − 2·76-s − 8·79-s + 3·83-s + ⋯ |
L(s) = 1 | − 4-s + 0.377·7-s − 1.10·13-s + 16-s − 0.727·17-s + 0.229·19-s − 0.625·23-s − 0.377·28-s − 1.67·29-s − 1.79·31-s + 0.657·37-s + 0.937·41-s − 0.152·43-s − 0.875·47-s + 1/7·49-s + 1.10·52-s − 1.23·53-s − 1.95·59-s + 1.66·61-s − 64-s − 0.244·67-s + 0.727·68-s + 0.712·71-s − 1.87·73-s − 0.229·76-s − 0.900·79-s + 0.329·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−13.50158423539662, −13.00816966997239, −12.77217333959893, −12.34006268867289, −11.61923111432442, −11.25797594033849, −10.81091958556659, −10.17654868647736, −9.611234361696056, −9.435444524241495, −8.876665593227748, −8.441774291577831, −7.636719392573735, −7.592411520910822, −7.022643612645272, −6.075398820058455, −5.827264422376153, −5.181046603219010, −4.665194468421556, −4.384125203927281, −3.617167325018791, −3.276224517034324, −2.306139624853560, −1.902964549394980, −1.142114970044820, 0, 0,
1.142114970044820, 1.902964549394980, 2.306139624853560, 3.276224517034324, 3.617167325018791, 4.384125203927281, 4.665194468421556, 5.181046603219010, 5.827264422376153, 6.075398820058455, 7.022643612645272, 7.592411520910822, 7.636719392573735, 8.441774291577831, 8.876665593227748, 9.435444524241495, 9.611234361696056, 10.17654868647736, 10.81091958556659, 11.25797594033849, 11.61923111432442, 12.34006268867289, 12.77217333959893, 13.00816966997239, 13.50158423539662