| L(s) = 1 | − 2-s + 4-s − 10·5-s − 14·7-s − 9·8-s + 10·10-s + 22·11-s − 22·13-s + 14·14-s − 47·16-s − 116·17-s + 102·19-s − 10·20-s − 22·22-s − 260·23-s + 75·25-s + 22·26-s − 14·28-s + 196·29-s + 150·31-s + 103·32-s + 116·34-s + 140·35-s − 96·37-s − 102·38-s + 90·40-s + 176·41-s + ⋯ |
| L(s) = 1 | − 0.353·2-s + 1/8·4-s − 0.894·5-s − 0.755·7-s − 0.397·8-s + 0.316·10-s + 0.603·11-s − 0.469·13-s + 0.267·14-s − 0.734·16-s − 1.65·17-s + 1.23·19-s − 0.111·20-s − 0.213·22-s − 2.35·23-s + 3/5·25-s + 0.165·26-s − 0.0944·28-s + 1.25·29-s + 0.869·31-s + 0.568·32-s + 0.585·34-s + 0.676·35-s − 0.426·37-s − 0.435·38-s + 0.355·40-s + 0.670·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + T + p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 p T + 2718 T^{2} - 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 22 T + 4450 T^{2} + 22 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 116 T + 9030 T^{2} + 116 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 102 T + 13134 T^{2} - 102 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 260 T + 34734 T^{2} + 260 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 196 T + 20942 T^{2} - 196 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 150 T + 36542 T^{2} - 150 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 96 T + 82550 T^{2} + 96 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 176 T - 16914 T^{2} - 176 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 p T + 171958 T^{2} + 8 p^{4} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 560 T + 248606 T^{2} + 560 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 326 T + 204138 T^{2} + 326 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 844 T + 474182 T^{2} - 844 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 204 T + 455006 T^{2} + 204 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 104 T + 537670 T^{2} + 104 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1670 T + 1384382 T^{2} + 1670 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 386 T + 152218 T^{2} + 386 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 888 T + 1007454 T^{2} + 888 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 928 T + 600710 T^{2} + 928 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 588 T + 1495334 T^{2} + 588 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 522 T + 291282 T^{2} - 522 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−11.20852434211860158070550292526, −10.34301194838350969192057189641, −9.912917910576036660002778366986, −9.813598143111122291097187789564, −8.941796239597540146908615809167, −8.733023208634280286081840199193, −8.220907487845438659299340479149, −7.65912481459610089706425179296, −7.08460507343796884953169348500, −6.57599935206256681515534697106, −6.34244154936345227806499879321, −5.62836816283299934943447396523, −4.56804878502591952579584650807, −4.48637974831833031466241011187, −3.68410070749499476586243317733, −2.99071596766604524419226575158, −2.39909018157542419291794056314, −1.38628743103439624835248470911, 0, 0,
1.38628743103439624835248470911, 2.39909018157542419291794056314, 2.99071596766604524419226575158, 3.68410070749499476586243317733, 4.48637974831833031466241011187, 4.56804878502591952579584650807, 5.62836816283299934943447396523, 6.34244154936345227806499879321, 6.57599935206256681515534697106, 7.08460507343796884953169348500, 7.65912481459610089706425179296, 8.220907487845438659299340479149, 8.733023208634280286081840199193, 8.941796239597540146908615809167, 9.813598143111122291097187789564, 9.912917910576036660002778366986, 10.34301194838350969192057189641, 11.20852434211860158070550292526
