| L(s) = 1 | − 4-s + 5·9-s + 6·11-s + 16-s + 14·19-s + 12·29-s − 8·31-s − 5·36-s − 18·41-s − 6·44-s − 49-s − 24·59-s − 20·61-s − 64-s + 12·71-s − 14·76-s − 28·79-s + 16·81-s + 30·89-s + 30·99-s − 28·109-s − 12·116-s + 5·121-s + 8·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 5/3·9-s + 1.80·11-s + 1/4·16-s + 3.21·19-s + 2.22·29-s − 1.43·31-s − 5/6·36-s − 2.81·41-s − 0.904·44-s − 1/7·49-s − 3.12·59-s − 2.56·61-s − 1/8·64-s + 1.42·71-s − 1.60·76-s − 3.15·79-s + 16/9·81-s + 3.17·89-s + 3.01·99-s − 2.68·109-s − 1.11·116-s + 5/11·121-s + 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.977753131\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.977753131\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 121 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−11.92881269295989137345621351552, −11.43542080794687841361161320888, −10.47171892529940308758171697404, −10.47158512922332734982283242374, −9.579009428886797063318045648029, −9.568685522922711961902715967641, −9.137854964217652677403368906993, −8.585608861462356437225398783181, −7.66578944834140103048419730306, −7.64782974353942160188456327130, −6.81224159074683869205437025071, −6.67972868613410665024264603752, −5.92173721221202844095837243996, −5.12974229193952936245223353764, −4.75532718895897824441470639549, −4.22140969165292187868312288693, −3.38090030408863777519563102748, −3.22476268795815471130630124211, −1.39931523367926616697580766515, −1.38643737698553398363216844504,
1.38643737698553398363216844504, 1.39931523367926616697580766515, 3.22476268795815471130630124211, 3.38090030408863777519563102748, 4.22140969165292187868312288693, 4.75532718895897824441470639549, 5.12974229193952936245223353764, 5.92173721221202844095837243996, 6.67972868613410665024264603752, 6.81224159074683869205437025071, 7.64782974353942160188456327130, 7.66578944834140103048419730306, 8.585608861462356437225398783181, 9.137854964217652677403368906993, 9.568685522922711961902715967641, 9.579009428886797063318045648029, 10.47158512922332734982283242374, 10.47171892529940308758171697404, 11.43542080794687841361161320888, 11.92881269295989137345621351552
