| L(s) = 1 | + 2·5-s + 4·9-s + 6·11-s + 12·19-s + 3·25-s + 4·37-s + 24·43-s + 8·45-s − 14·49-s − 12·53-s + 12·55-s − 12·79-s + 7·81-s + 24·95-s + 20·97-s + 24·99-s − 24·107-s + 12·113-s + 25·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 4/3·9-s + 1.80·11-s + 2.75·19-s + 3/5·25-s + 0.657·37-s + 3.65·43-s + 1.19·45-s − 2·49-s − 1.64·53-s + 1.61·55-s − 1.35·79-s + 7/9·81-s + 2.46·95-s + 2.03·97-s + 2.41·99-s − 2.32·107-s + 1.12·113-s + 2.27·121-s + 0.357·125-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12390400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12390400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.158500500\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.158500500\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 116 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−9.043198468985565589488910016549, −8.406325306563265521345201088295, −7.78967575057676783366482183534, −7.69367740371030396197682898256, −7.18187394680856279798617302576, −7.01500942983911510359938959028, −6.43213469118564884042929357754, −6.26657809058434746842727422661, −5.65540480461505028682953548893, −5.58816322391410379358100014558, −4.85155753735752241737102293311, −4.54960030637897651010073842077, −4.19950029990844915510630643834, −3.71110093936048037926549071352, −3.19040331322851124521240211481, −2.91318055454711279554261346901, −2.12752281685319600276413929971, −1.61873366022386686966266837865, −1.13165032415931012504338377982, −0.924367727282552155481263697335,
0.924367727282552155481263697335, 1.13165032415931012504338377982, 1.61873366022386686966266837865, 2.12752281685319600276413929971, 2.91318055454711279554261346901, 3.19040331322851124521240211481, 3.71110093936048037926549071352, 4.19950029990844915510630643834, 4.54960030637897651010073842077, 4.85155753735752241737102293311, 5.58816322391410379358100014558, 5.65540480461505028682953548893, 6.26657809058434746842727422661, 6.43213469118564884042929357754, 7.01500942983911510359938959028, 7.18187394680856279798617302576, 7.69367740371030396197682898256, 7.78967575057676783366482183534, 8.406325306563265521345201088295, 9.043198468985565589488910016549
