| L(s) = 1 | + 2·2-s − 4-s + 8·7-s − 8·8-s − 6·9-s + 8·13-s + 16·14-s − 7·16-s − 12·18-s + 2·25-s + 16·26-s − 8·28-s + 14·32-s + 6·36-s + 8·37-s − 24·41-s + 8·43-s + 8·47-s + 34·49-s + 4·50-s − 8·52-s − 64·56-s − 48·63-s + 35·64-s + 48·72-s + 16·74-s − 8·79-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1/2·4-s + 3.02·7-s − 2.82·8-s − 2·9-s + 2.21·13-s + 4.27·14-s − 7/4·16-s − 2.82·18-s + 2/5·25-s + 3.13·26-s − 1.51·28-s + 2.47·32-s + 36-s + 1.31·37-s − 3.74·41-s + 1.21·43-s + 1.16·47-s + 34/7·49-s + 0.565·50-s − 1.10·52-s − 8.55·56-s − 6.04·63-s + 35/8·64-s + 5.65·72-s + 1.85·74-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 113569 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 113569 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.748771867\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.748771867\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 337 | $C_2$ | \( 1 + 14 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 36 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 36 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 140 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
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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
−11.70740020096550720385501259744, −11.48295159801118849980109180965, −11.14793964859192555754674028391, −10.74731988148015954094747055229, −10.05529420299817131044268784770, −9.014907172834271248127853224637, −8.729950701673612260764113581547, −8.674024342809815630169868836506, −8.036764620071368126257527627601, −7.970993110466249082264355356383, −6.72444933825401270334965867679, −5.91419853498388762459377960963, −5.72824505808447978453387496334, −5.32448829835370133469892164473, −4.66285618978251705776087855087, −4.51001966647882472257315803589, −3.62584723147185061483852731165, −3.24401518442648583211124489149, −2.22291345220254145009396503249, −1.06457803857429248733392161551,
1.06457803857429248733392161551, 2.22291345220254145009396503249, 3.24401518442648583211124489149, 3.62584723147185061483852731165, 4.51001966647882472257315803589, 4.66285618978251705776087855087, 5.32448829835370133469892164473, 5.72824505808447978453387496334, 5.91419853498388762459377960963, 6.72444933825401270334965867679, 7.970993110466249082264355356383, 8.036764620071368126257527627601, 8.674024342809815630169868836506, 8.729950701673612260764113581547, 9.014907172834271248127853224637, 10.05529420299817131044268784770, 10.74731988148015954094747055229, 11.14793964859192555754674028391, 11.48295159801118849980109180965, 11.70740020096550720385501259744
