| L(s) = 1 | + 29·9-s − 30·11-s + 308·19-s − 6·29-s − 656·31-s + 192·41-s − 49·49-s + 792·59-s − 1.23e3·61-s − 96·71-s − 1.31e3·79-s + 112·81-s − 2.04e3·89-s − 870·99-s − 1.11e3·101-s + 1.11e3·109-s − 1.98e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.22e3·169-s + ⋯ |
| L(s) = 1 | + 1.07·9-s − 0.822·11-s + 3.71·19-s − 0.0384·29-s − 3.80·31-s + 0.731·41-s − 1/7·49-s + 1.74·59-s − 2.58·61-s − 0.160·71-s − 1.87·79-s + 0.153·81-s − 2.42·89-s − 0.883·99-s − 1.09·101-s + 0.982·109-s − 1.49·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 1.92·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.343835589\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.343835589\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 29 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 15 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 p^{2} T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9097 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 154 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 10262 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 328 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 36790 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 96 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 141058 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 205045 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 240154 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 396 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 616 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 513910 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 48 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 674350 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 659 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1053574 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 1020 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1785745 T^{2} + p^{6} T^{4} \) |
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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
−10.24596887564137981180999711926, −9.725462543791945523201240284061, −9.534447164062911128104930676243, −9.063885017807129307522472506349, −8.702992688677304931311267145591, −7.74615508955995035055934564956, −7.62458483619559156246944325189, −7.29478386672381562205958615587, −7.06877968303343717516942919498, −6.22346483647754092085761121458, −5.55535829090111120847579173884, −5.26756518755821532965536973033, −5.12141224720612801543675349564, −4.04684454966050300423445733432, −3.87805014364890377173652343770, −3.04385476804507960785443769536, −2.78930611644903739218298439148, −1.55943915766731846150829802049, −1.47333023662625535067172145300, −0.43813688769191033024567202800,
0.43813688769191033024567202800, 1.47333023662625535067172145300, 1.55943915766731846150829802049, 2.78930611644903739218298439148, 3.04385476804507960785443769536, 3.87805014364890377173652343770, 4.04684454966050300423445733432, 5.12141224720612801543675349564, 5.26756518755821532965536973033, 5.55535829090111120847579173884, 6.22346483647754092085761121458, 7.06877968303343717516942919498, 7.29478386672381562205958615587, 7.62458483619559156246944325189, 7.74615508955995035055934564956, 8.702992688677304931311267145591, 9.063885017807129307522472506349, 9.534447164062911128104930676243, 9.725462543791945523201240284061, 10.24596887564137981180999711926
