| L(s) = 1 | + 2·2-s − 3·3-s + 3·4-s + 5-s − 6·6-s + 2·7-s + 4·8-s + 2·9-s + 2·10-s + 7·11-s − 9·12-s − 6·13-s + 4·14-s − 3·15-s + 5·16-s − 4·17-s + 4·18-s + 3·20-s − 6·21-s + 14·22-s + 2·23-s − 12·24-s + 2·25-s − 12·26-s + 6·27-s + 6·28-s + 11·29-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.73·3-s + 3/2·4-s + 0.447·5-s − 2.44·6-s + 0.755·7-s + 1.41·8-s + 2/3·9-s + 0.632·10-s + 2.11·11-s − 2.59·12-s − 1.66·13-s + 1.06·14-s − 0.774·15-s + 5/4·16-s − 0.970·17-s + 0.942·18-s + 0.670·20-s − 1.30·21-s + 2.98·22-s + 0.417·23-s − 2.44·24-s + 2/5·25-s − 2.35·26-s + 1.15·27-s + 1.13·28-s + 2.04·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25542916 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25542916 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.859308882\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.859308882\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 7 T + 3 p T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 11 T + 77 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 58 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 3 T + 65 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 5 T + 27 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 13 T + 127 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + T + 83 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 25 T + 261 T^{2} - 25 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 11 T + 137 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 11 T + 91 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 3 T - 67 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 10 T + 166 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 11 T + 127 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 5 T + 183 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + T + 183 T^{2} + p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.430918513188159061896563010300, −7.990484800497889240384824630941, −7.21433769115620377129407222380, −7.03883930306591710024813396994, −6.74566587582782840978527525201, −6.72305413844112609139115057050, −6.02205332902358180495311574204, −5.88481210995094417572137475359, −5.39751163031909061757464077215, −5.16943988569007445921109112002, −4.79071729405998268668481335376, −4.60081715973733633038865698681, −4.06075393716654564195864050623, −3.79420894479465020591641029311, −3.16217422709710157698057640389, −2.58475834148330168464255290766, −2.21483928580706306924034669323, −1.74505526392658759722624175652, −1.06267392337163156696937956249, −0.57483547744143112307305337657,
0.57483547744143112307305337657, 1.06267392337163156696937956249, 1.74505526392658759722624175652, 2.21483928580706306924034669323, 2.58475834148330168464255290766, 3.16217422709710157698057640389, 3.79420894479465020591641029311, 4.06075393716654564195864050623, 4.60081715973733633038865698681, 4.79071729405998268668481335376, 5.16943988569007445921109112002, 5.39751163031909061757464077215, 5.88481210995094417572137475359, 6.02205332902358180495311574204, 6.72305413844112609139115057050, 6.74566587582782840978527525201, 7.03883930306591710024813396994, 7.21433769115620377129407222380, 7.990484800497889240384824630941, 8.430918513188159061896563010300
