| L(s) = 1 | + 10·5-s + 50·9-s + 56·11-s − 120·19-s − 25·25-s + 180·29-s − 256·31-s + 484·41-s + 500·45-s + 10·49-s + 560·55-s − 40·59-s − 1.08e3·61-s − 2.25e3·71-s + 1.44e3·79-s + 1.77e3·81-s + 980·89-s − 1.20e3·95-s + 2.80e3·99-s + 1.15e3·101-s + 740·109-s − 310·121-s − 1.50e3·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.85·9-s + 1.53·11-s − 1.44·19-s − 1/5·25-s + 1.15·29-s − 1.48·31-s + 1.84·41-s + 1.65·45-s + 0.0291·49-s + 1.37·55-s − 0.0882·59-s − 2.27·61-s − 3.77·71-s + 2.05·79-s + 2.42·81-s + 1.16·89-s − 1.29·95-s + 2.84·99-s + 1.13·101-s + 0.650·109-s − 0.232·121-s − 1.07·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.151778207\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.151778207\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 - 2 p T + p^{3} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 50 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 28 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4250 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 5730 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 60 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 20970 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 90 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 128 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 45610 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 242 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 27970 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 156570 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 286090 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 20 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 542 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 413170 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 1128 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 378610 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 720 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 915090 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 490 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 294590 T^{2} + p^{6} T^{4} \) |
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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
−11.44882135083684824148929030341, −10.65293564668183674781522240351, −10.57414412333405307026209586741, −10.11556071257465796326139075582, −9.436179251195188422269726643836, −9.119870743668240181451128152218, −8.995261770823402169029000634345, −8.010867569235655360101214641531, −7.56716214727069016926041410952, −7.00129800582575677014624815039, −6.55304683624049432613562631405, −6.13574437147212438980891751340, −5.68297296987992963923648045740, −4.61104182251339335596954819002, −4.39864792936923166987060598602, −3.89482963891029657179100245135, −3.02336676388724617027351559708, −1.84391091884897757701568395479, −1.75232921695408700796812412207, −0.77660257682822605928917914978,
0.77660257682822605928917914978, 1.75232921695408700796812412207, 1.84391091884897757701568395479, 3.02336676388724617027351559708, 3.89482963891029657179100245135, 4.39864792936923166987060598602, 4.61104182251339335596954819002, 5.68297296987992963923648045740, 6.13574437147212438980891751340, 6.55304683624049432613562631405, 7.00129800582575677014624815039, 7.56716214727069016926041410952, 8.010867569235655360101214641531, 8.995261770823402169029000634345, 9.119870743668240181451128152218, 9.436179251195188422269726643836, 10.11556071257465796326139075582, 10.57414412333405307026209586741, 10.65293564668183674781522240351, 11.44882135083684824148929030341
