L(s) = 1 | − 2-s + 4-s + 4·5-s − 8-s − 4·10-s + 12·13-s + 16-s − 4·17-s + 4·20-s + 2·25-s − 12·26-s + 4·29-s − 32-s + 4·34-s − 20·37-s − 4·40-s + 12·41-s + 49-s − 2·50-s + 12·52-s − 12·53-s − 4·58-s + 12·61-s + 64-s + 48·65-s − 4·68-s + 20·73-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.78·5-s − 0.353·8-s − 1.26·10-s + 3.32·13-s + 1/4·16-s − 0.970·17-s + 0.894·20-s + 2/5·25-s − 2.35·26-s + 0.742·29-s − 0.176·32-s + 0.685·34-s − 3.28·37-s − 0.632·40-s + 1.87·41-s + 1/7·49-s − 0.282·50-s + 1.66·52-s − 1.64·53-s − 0.525·58-s + 1.53·61-s + 1/8·64-s + 5.95·65-s − 0.485·68-s + 2.34·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 127008 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127008 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.826911554\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.826911554\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 3 | | \( 1 \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−9.164662289745378790951914691450, −9.085422624242741773351689713512, −8.427817591055194605299416158333, −8.337652051623380123011624806396, −7.46998199559376075566652373057, −6.58813712157544804919343144385, −6.48878700424638503015501865218, −6.01899647743800981343957937964, −5.58110052520146908580921209772, −4.96240850233932644050318806427, −3.83863171110134241641513411613, −3.57539886636491016875562737772, −2.47676327970158091646809943718, −1.79365373540820646322667624458, −1.22168166913217334012046287130,
1.22168166913217334012046287130, 1.79365373540820646322667624458, 2.47676327970158091646809943718, 3.57539886636491016875562737772, 3.83863171110134241641513411613, 4.96240850233932644050318806427, 5.58110052520146908580921209772, 6.01899647743800981343957937964, 6.48878700424638503015501865218, 6.58813712157544804919343144385, 7.46998199559376075566652373057, 8.337652051623380123011624806396, 8.427817591055194605299416158333, 9.085422624242741773351689713512, 9.164662289745378790951914691450