| L(s) = 1 | − 4·2-s + 8·4-s − 8·8-s + 4·11-s − 4·16-s − 16·22-s − 6·25-s − 8·29-s + 32·32-s + 6·37-s + 10·43-s + 32·44-s + 24·50-s − 24·53-s + 32·58-s − 64·64-s − 10·67-s + 12·71-s − 24·74-s − 2·79-s − 40·86-s − 32·88-s − 48·100-s + 96·106-s + 16·107-s + 18·109-s − 20·113-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 4·4-s − 2.82·8-s + 1.20·11-s − 16-s − 3.41·22-s − 6/5·25-s − 1.48·29-s + 5.65·32-s + 0.986·37-s + 1.52·43-s + 4.82·44-s + 3.39·50-s − 3.29·53-s + 4.20·58-s − 8·64-s − 1.22·67-s + 1.42·71-s − 2.78·74-s − 0.225·79-s − 4.31·86-s − 3.41·88-s − 4.79·100-s + 9.32·106-s + 1.54·107-s + 1.72·109-s − 1.88·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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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
−8.970613305827843910423804944183, −8.583706845901838358159961836152, −7.920026489093004352764513852408, −7.67312246287158311998952783143, −7.42251681884249141719743547364, −6.65130832285966559117996014878, −6.33310252534575837626677109971, −5.72037257717675019727399870073, −4.73946994522750223509171420340, −4.23783192793666623809357478808, −3.54901712994340686137812719330, −2.43327032489084752763662196914, −1.75196450155362895800400842510, −1.15383453307155328641933378041, 0,
1.15383453307155328641933378041, 1.75196450155362895800400842510, 2.43327032489084752763662196914, 3.54901712994340686137812719330, 4.23783192793666623809357478808, 4.73946994522750223509171420340, 5.72037257717675019727399870073, 6.33310252534575837626677109971, 6.65130832285966559117996014878, 7.42251681884249141719743547364, 7.67312246287158311998952783143, 7.920026489093004352764513852408, 8.583706845901838358159961836152, 8.970613305827843910423804944183
