| L(s) = 1 | + 4·2-s + 4-s + 10·5-s − 14·7-s − 24·8-s + 40·10-s + 92·11-s + 8·13-s − 56·14-s − 47·16-s + 44·17-s − 108·19-s + 10·20-s + 368·22-s + 320·23-s + 75·25-s + 32·26-s − 14·28-s + 236·29-s − 60·31-s − 52·32-s + 176·34-s − 140·35-s + 204·37-s − 432·38-s − 240·40-s − 44·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/8·4-s + 0.894·5-s − 0.755·7-s − 1.06·8-s + 1.26·10-s + 2.52·11-s + 0.170·13-s − 1.06·14-s − 0.734·16-s + 0.627·17-s − 1.30·19-s + 0.111·20-s + 3.56·22-s + 2.90·23-s + 3/5·25-s + 0.241·26-s − 0.0944·28-s + 1.51·29-s − 0.347·31-s − 0.287·32-s + 0.887·34-s − 0.676·35-s + 0.906·37-s − 1.84·38-s − 0.948·40-s − 0.167·41-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
| \(L(2)\) |
\(\approx\) |
\(6.565178106\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.565178106\) |
| \(L(\frac{5}{2})\) |
|
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 | $D_{4}$ | \( 1 - p^{2} T + 15 T^{2} - p^{5} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 92 T + 4758 T^{2} - 92 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 8 T - 2810 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 44 T + 630 T^{2} - 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 108 T + 13254 T^{2} + 108 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 320 T + 47934 T^{2} - 320 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 236 T + 61982 T^{2} - 236 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 60 T + 8462 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 204 T + 108830 T^{2} - 204 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 44 T + 106326 T^{2} + 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 136 T + 81718 T^{2} - 136 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 400 T + 106526 T^{2} + 400 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 16 T + 260838 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 464 T + 458102 T^{2} - 464 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 684 T + 535646 T^{2} + 684 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 736 T + 602470 T^{2} - 736 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 740 T + 757502 T^{2} - 740 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 424 T + 748558 T^{2} - 424 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 408 T - 143586 T^{2} + 408 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 608 T + 1200710 T^{2} + 608 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 1332 T + 1790774 T^{2} - 1332 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2448 T + 3286542 T^{2} + 2448 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−11.47358152864396177753547895408, −11.20677640210453398427372616145, −10.61647835406717912417548105543, −9.975343523743146639571214040275, −9.360436287965283379255783694051, −9.345759010336154905613799357586, −8.723143593667288955626460298263, −8.431989026082797058890740456587, −7.30745530774722496150373155788, −6.71172219518177827087069410759, −6.37903628467641081511448377046, −6.20556051782324203043552656359, −5.23026954342136069670380104025, −5.02795655889557279370790780734, −4.22450796082048610629875225983, −3.90168472031826049292867180971, −3.23124910964062948756853364696, −2.67729233291198040836624339395, −1.45943654222981331992284209107, −0.841711399501427925331087035863,
0.841711399501427925331087035863, 1.45943654222981331992284209107, 2.67729233291198040836624339395, 3.23124910964062948756853364696, 3.90168472031826049292867180971, 4.22450796082048610629875225983, 5.02795655889557279370790780734, 5.23026954342136069670380104025, 6.20556051782324203043552656359, 6.37903628467641081511448377046, 6.71172219518177827087069410759, 7.30745530774722496150373155788, 8.431989026082797058890740456587, 8.723143593667288955626460298263, 9.345759010336154905613799357586, 9.360436287965283379255783694051, 9.975343523743146639571214040275, 10.61647835406717912417548105543, 11.20677640210453398427372616145, 11.47358152864396177753547895408
