| L(s) = 1 | + 3·4-s + 6·9-s − 2·11-s + 5·16-s + 10·19-s + 10·29-s − 4·31-s + 18·36-s − 10·41-s − 6·44-s + 13·49-s − 16·59-s − 16·61-s + 3·64-s − 20·71-s + 30·76-s + 6·79-s + 27·81-s − 20·89-s − 12·99-s + 20·101-s + 8·109-s + 30·116-s − 19·121-s − 12·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 3/2·4-s + 2·9-s − 0.603·11-s + 5/4·16-s + 2.29·19-s + 1.85·29-s − 0.718·31-s + 3·36-s − 1.56·41-s − 0.904·44-s + 13/7·49-s − 2.08·59-s − 2.04·61-s + 3/8·64-s − 2.37·71-s + 3.44·76-s + 0.675·79-s + 3·81-s − 2.11·89-s − 1.20·99-s + 1.99·101-s + 0.766·109-s + 2.78·116-s − 1.72·121-s − 1.07·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 330625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 330625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.414582879\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.414582879\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 23 | $C_2$ | \( 1 + T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 137 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 157 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.76183474404696328341362021599, −10.49241306328573514318982599361, −10.09150640252760371332045489212, −9.992030312332282372643971193716, −9.059712843247649867851277286611, −9.037131810762132087136238790310, −7.88440055362772400040473953656, −7.77819641929560309992472826725, −7.28444473954647255246262256191, −7.05757894677444735166958231908, −6.54980048175792380613431257080, −6.03915403854247655704753470226, −5.46805635250956920064535360545, −4.83412161539315577226130851698, −4.47370292965563960650316467620, −3.59539450393613545204777005093, −3.07626076952241090642390018840, −2.55263904077270303806880937967, −1.57368675459621306887632914558, −1.26745356576081384359765677173,
1.26745356576081384359765677173, 1.57368675459621306887632914558, 2.55263904077270303806880937967, 3.07626076952241090642390018840, 3.59539450393613545204777005093, 4.47370292965563960650316467620, 4.83412161539315577226130851698, 5.46805635250956920064535360545, 6.03915403854247655704753470226, 6.54980048175792380613431257080, 7.05757894677444735166958231908, 7.28444473954647255246262256191, 7.77819641929560309992472826725, 7.88440055362772400040473953656, 9.037131810762132087136238790310, 9.059712843247649867851277286611, 9.992030312332282372643971193716, 10.09150640252760371332045489212, 10.49241306328573514318982599361, 10.76183474404696328341362021599
