| L(s) = 1 | − 9-s + 4·29-s + 16·31-s − 12·41-s − 2·49-s + 16·59-s − 20·61-s − 16·71-s + 32·79-s + 81-s + 12·89-s + 12·101-s − 12·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 0.742·29-s + 2.87·31-s − 1.87·41-s − 2/7·49-s + 2.08·59-s − 2.56·61-s − 1.89·71-s + 3.60·79-s + 1/9·81-s + 1.27·89-s + 1.19·101-s − 1.14·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.126898862\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.126898862\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.099936068691839496470025577361, −8.726219100359906344886230884722, −8.323044228833822369117650730454, −8.126287749808521210079267879419, −7.67195713806345899523691634576, −7.31124558161405289740090343325, −6.55951524465885967112661411145, −6.54688622667935621416527552001, −6.23308307035099267761853194677, −5.62386951348814670831543375902, −5.13859536078865449473930271550, −4.74966005830495901085915361792, −4.53469461670118861338206993111, −3.84083195096377752945549591447, −3.41867939964718354915968280596, −2.88394534277457809963563669863, −2.55645597279155231404210347734, −1.88090725973862036726533641226, −1.21314626643241096783196699524, −0.53411258984026040818059524128,
0.53411258984026040818059524128, 1.21314626643241096783196699524, 1.88090725973862036726533641226, 2.55645597279155231404210347734, 2.88394534277457809963563669863, 3.41867939964718354915968280596, 3.84083195096377752945549591447, 4.53469461670118861338206993111, 4.74966005830495901085915361792, 5.13859536078865449473930271550, 5.62386951348814670831543375902, 6.23308307035099267761853194677, 6.54688622667935621416527552001, 6.55951524465885967112661411145, 7.31124558161405289740090343325, 7.67195713806345899523691634576, 8.126287749808521210079267879419, 8.323044228833822369117650730454, 8.726219100359906344886230884722, 9.099936068691839496470025577361
