| L(s) = 1 | + 2·2-s − 4·3-s + 10·5-s − 8·6-s − 23·7-s − 8·8-s + 27·9-s + 20·10-s + 57·11-s + 65·13-s − 46·14-s − 40·15-s − 16·16-s − 48·17-s + 54·18-s − 5·19-s + 92·21-s + 114·22-s + 180·23-s + 32·24-s + 75·25-s + 130·26-s − 260·27-s − 120·29-s − 80·30-s − 68·31-s − 228·33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.769·3-s + 0.894·5-s − 0.544·6-s − 1.24·7-s − 0.353·8-s + 9-s + 0.632·10-s + 1.56·11-s + 1.38·13-s − 0.878·14-s − 0.688·15-s − 1/4·16-s − 0.684·17-s + 0.707·18-s − 0.0603·19-s + 0.956·21-s + 1.10·22-s + 1.63·23-s + 0.272·24-s + 3/5·25-s + 0.980·26-s − 1.85·27-s − 0.768·29-s − 0.486·30-s − 0.393·31-s − 1.20·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16900 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.684223945\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.684223945\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 5 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 13 | $C_2$ | \( 1 - 5 p T + p^{3} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T - 11 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 23 T + 186 T^{2} + 23 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 57 T + 1918 T^{2} - 57 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 48 T - 2609 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 5 T - 6834 T^{2} + 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 180 T + 20233 T^{2} - 180 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 120 T - 9989 T^{2} + 120 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 34 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 227 T + 876 T^{2} + 227 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 68885 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 214 T - 33711 T^{2} - 214 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 549 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 537 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 300 T - 115379 T^{2} - 300 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 322 T - 123297 T^{2} - 322 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 772 T + 295221 T^{2} - 772 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 894 T + 441325 T^{2} - 894 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 272 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 454 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 1068 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 681 T - 241208 T^{2} + 681 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 1100 T + 297327 T^{2} + 1100 p^{3} T^{3} + p^{6} T^{4} \) |
| 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
−12.89461169935898225616216371801, −12.73908115642514526057660246057, −12.40865183980290750322979168995, −11.43197101798193015929860728927, −11.11821788385570450073264642642, −10.76230641802309785108426634483, −9.788362856959135162968588712389, −9.569693231614294007940982437530, −8.994766907182463086260911300857, −8.647028526777869761665566763904, −7.25902530047983837218610816173, −6.81804526585483156637465294991, −6.46684337682057969541563509546, −5.64778361295136367591919095661, −5.61278106449379455300300076894, −4.32537942237742667542397640352, −3.92026347377917947574781683975, −3.18209893424936274385418290125, −1.85300770565945585019181488822, −0.841612841887339250914418590528,
0.841612841887339250914418590528, 1.85300770565945585019181488822, 3.18209893424936274385418290125, 3.92026347377917947574781683975, 4.32537942237742667542397640352, 5.61278106449379455300300076894, 5.64778361295136367591919095661, 6.46684337682057969541563509546, 6.81804526585483156637465294991, 7.25902530047983837218610816173, 8.647028526777869761665566763904, 8.994766907182463086260911300857, 9.569693231614294007940982437530, 9.788362856959135162968588712389, 10.76230641802309785108426634483, 11.11821788385570450073264642642, 11.43197101798193015929860728927, 12.40865183980290750322979168995, 12.73908115642514526057660246057, 12.89461169935898225616216371801
