| L(s) = 1 | − 8·5-s + 40·13-s + 20·17-s − 2·25-s − 40·29-s + 40·37-s − 60·41-s − 30·49-s + 120·53-s − 56·61-s − 320·65-s + 20·73-s − 160·85-s + 44·89-s + 300·97-s + 280·101-s + 136·109-s + 380·113-s + 42·121-s + 344·125-s + 127-s + 131-s + 137-s + 139-s + 320·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 8/5·5-s + 3.07·13-s + 1.17·17-s − 0.0799·25-s − 1.37·29-s + 1.08·37-s − 1.46·41-s − 0.612·49-s + 2.26·53-s − 0.918·61-s − 4.92·65-s + 0.273·73-s − 1.88·85-s + 0.494·89-s + 3.09·97-s + 2.77·101-s + 1.24·109-s + 3.36·113-s + 0.347·121-s + 2.75·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 2.20·145-s + 0.00671·149-s + 0.00662·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.002255927\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.002255927\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + 4 T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 30 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 42 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 20 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 522 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 930 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 20 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 20 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 30 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 3690 T^{2} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 190 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 60 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 5162 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 28 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 2250 T^{2} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 6882 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 318 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 13130 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 150 T + p^{2} T^{2} )^{2} \) |
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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
−8.751828618565635932371323938952, −8.667599103170719997838009395825, −8.261229264745369855686063292908, −7.86381253846379394250593175967, −7.44051169026616000764895701148, −7.41852044335674216715256851285, −6.68112255914800431692252777346, −6.18820899048238082691885470898, −5.84896518757649837806522123420, −5.74513256137580035355323991968, −4.97871121563969503683805422051, −4.47979563917095382918232565682, −4.01757048219578845455249827276, −3.64892229327472694461089165781, −3.34904623498636734048225918846, −3.24021065722272968572757434491, −1.99948542631572431683018170011, −1.71901595482786563279102616159, −0.76767039936684423367409329050, −0.63448632540110217153682705338,
0.63448632540110217153682705338, 0.76767039936684423367409329050, 1.71901595482786563279102616159, 1.99948542631572431683018170011, 3.24021065722272968572757434491, 3.34904623498636734048225918846, 3.64892229327472694461089165781, 4.01757048219578845455249827276, 4.47979563917095382918232565682, 4.97871121563969503683805422051, 5.74513256137580035355323991968, 5.84896518757649837806522123420, 6.18820899048238082691885470898, 6.68112255914800431692252777346, 7.41852044335674216715256851285, 7.44051169026616000764895701148, 7.86381253846379394250593175967, 8.261229264745369855686063292908, 8.667599103170719997838009395825, 8.751828618565635932371323938952
