| L(s) = 1 | − 4·2-s + 6·3-s + 12·4-s + 18·5-s − 24·6-s + 4·7-s − 32·8-s + 27·9-s − 72·10-s + 60·11-s + 72·12-s − 58·13-s − 16·14-s + 108·15-s + 80·16-s + 96·17-s − 108·18-s + 216·20-s + 24·21-s − 240·22-s − 30·23-s − 192·24-s + 146·25-s + 232·26-s + 108·27-s + 48·28-s + 252·29-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s + 1.60·5-s − 1.63·6-s + 0.215·7-s − 1.41·8-s + 9-s − 2.27·10-s + 1.64·11-s + 1.73·12-s − 1.23·13-s − 0.305·14-s + 1.85·15-s + 5/4·16-s + 1.36·17-s − 1.41·18-s + 2.41·20-s + 0.249·21-s − 2.32·22-s − 0.271·23-s − 1.63·24-s + 1.16·25-s + 1.74·26-s + 0.769·27-s + 0.323·28-s + 1.61·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4691556 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4691556 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.056077685\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.056077685\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 19 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 - 18 T + 178 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 60 T + 2950 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 58 T + 3858 T^{2} + 58 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 96 T + 11518 T^{2} - 96 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 30 T + 24406 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 252 T + 54862 T^{2} - 252 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 338 T + 69630 T^{2} - 338 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 158 T + 100050 T^{2} - 158 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 142 T^{2} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 248 T + 164598 T^{2} + 248 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 810 T + 7514 p T^{2} + 810 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 84 T + 54718 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 504 T + 121750 T^{2} - 504 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 580 T + 241854 T^{2} - 580 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 140 T + 502998 T^{2} - 140 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 480 T + 420910 T^{2} - 480 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 616 T + 842910 T^{2} - 616 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 202 T + 763566 T^{2} + 202 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 552 T + 1158550 T^{2} + 552 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 924 T + 1620934 T^{2} - 924 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 40 T - 162642 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.901695475542930537967913926950, −8.513682831714983953618692787713, −8.263930751994015957708677127117, −7.990201507508828905927715428142, −7.52457468463176172625190547867, −6.92632442676811522889070157955, −6.70241240445522337611194305351, −6.39652339728826062792718207563, −6.03402463518103023936250962771, −5.44427029993029459387067026619, −4.77303115804363250798731202363, −4.70249218859681427884184176907, −3.76511192912798670741053928931, −3.36796895533978904688034319744, −2.77138405209028466936310513368, −2.52161133166284687446011819579, −1.85051222610797864282310473991, −1.66054860614316583459314225871, −0.978742042887023599296780473539, −0.71376614259200892827981122893,
0.71376614259200892827981122893, 0.978742042887023599296780473539, 1.66054860614316583459314225871, 1.85051222610797864282310473991, 2.52161133166284687446011819579, 2.77138405209028466936310513368, 3.36796895533978904688034319744, 3.76511192912798670741053928931, 4.70249218859681427884184176907, 4.77303115804363250798731202363, 5.44427029993029459387067026619, 6.03402463518103023936250962771, 6.39652339728826062792718207563, 6.70241240445522337611194305351, 6.92632442676811522889070157955, 7.52457468463176172625190547867, 7.990201507508828905927715428142, 8.263930751994015957708677127117, 8.513682831714983953618692787713, 8.901695475542930537967913926950
