| L(s) = 1 | + 2.41·2-s + 3.82·4-s + 2·7-s + 4.41·8-s − 11-s + 1.17·13-s + 4.82·14-s + 2.99·16-s + 6.82·17-s − 2.41·22-s − 2.82·23-s + 2.82·26-s + 7.65·28-s + 3.65·29-s − 1.58·32-s + 16.4·34-s + 7.65·37-s − 6·41-s + 6·43-s − 3.82·44-s − 6.82·46-s + 2.82·47-s − 3·49-s + 4.48·52-s + 11.6·53-s + 8.82·56-s + 8.82·58-s + ⋯ |
| L(s) = 1 | + 1.70·2-s + 1.91·4-s + 0.755·7-s + 1.56·8-s − 0.301·11-s + 0.324·13-s + 1.29·14-s + 0.749·16-s + 1.65·17-s − 0.514·22-s − 0.589·23-s + 0.554·26-s + 1.44·28-s + 0.679·29-s − 0.280·32-s + 2.82·34-s + 1.25·37-s − 0.937·41-s + 0.914·43-s − 0.577·44-s − 1.00·46-s + 0.412·47-s − 0.428·49-s + 0.621·52-s + 1.60·53-s + 1.17·56-s + 1.15·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.699684376\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.699684376\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 2.41T + 2T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 13 | \( 1 - 1.17T + 13T^{2} \) |
| 17 | \( 1 - 6.82T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 - 3.65T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 7.65T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 + 1.65T + 59T^{2} \) |
| 61 | \( 1 + 9.31T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 - 1.17T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 + 3.65T + 97T^{2} \) |
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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
−8.775792573652341552029370576849, −7.84436201939499780430693494726, −7.32837030396739543953959643528, −6.19136387220385288591921419836, −5.71026623909183260703198512596, −4.92272567370070635036782669109, −4.23103054301320907439201112597, −3.36029176631802819053516109770, −2.52939341252480033016628415450, −1.35717830261851675732069691262,
1.35717830261851675732069691262, 2.52939341252480033016628415450, 3.36029176631802819053516109770, 4.23103054301320907439201112597, 4.92272567370070635036782669109, 5.71026623909183260703198512596, 6.19136387220385288591921419836, 7.32837030396739543953959643528, 7.84436201939499780430693494726, 8.775792573652341552029370576849
