L(s) = 1 | + (−0.815 − 0.578i)2-s + (0.329 + 0.944i)4-s + (0.904 − 0.426i)5-s + (0.277 − 0.960i)8-s + (−0.984 − 0.175i)10-s + (0.191 − 0.981i)11-s + (−0.371 − 0.928i)13-s + (−0.782 + 0.622i)16-s + (0.00551 + 0.999i)17-s + (−0.984 + 0.175i)19-s + (0.701 + 0.712i)20-s + (−0.724 + 0.689i)22-s + (0.899 + 0.436i)23-s + (0.635 − 0.771i)25-s + (−0.234 + 0.972i)26-s + ⋯ |
L(s) = 1 | + (−0.815 − 0.578i)2-s + (0.329 + 0.944i)4-s + (0.904 − 0.426i)5-s + (0.277 − 0.960i)8-s + (−0.984 − 0.175i)10-s + (0.191 − 0.981i)11-s + (−0.371 − 0.928i)13-s + (−0.782 + 0.622i)16-s + (0.00551 + 0.999i)17-s + (−0.984 + 0.175i)19-s + (0.701 + 0.712i)20-s + (−0.724 + 0.689i)22-s + (0.899 + 0.436i)23-s + (0.635 − 0.771i)25-s + (−0.234 + 0.972i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.311 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.311 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1094852594 + 0.1511214577i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1094852594 + 0.1511214577i\) |
\(L(1)\) |
\(\approx\) |
\(0.6680605898 - 0.2029275335i\) |
\(L(1)\) |
\(\approx\) |
\(0.6680605898 - 0.2029275335i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 191 | \( 1 \) |
good | 2 | \( 1 + (-0.815 - 0.578i)T \) |
| 5 | \( 1 + (0.904 - 0.426i)T \) |
| 11 | \( 1 + (0.191 - 0.981i)T \) |
| 13 | \( 1 + (-0.371 - 0.928i)T \) |
| 17 | \( 1 + (0.00551 + 0.999i)T \) |
| 19 | \( 1 + (-0.984 + 0.175i)T \) |
| 23 | \( 1 + (0.899 + 0.436i)T \) |
| 29 | \( 1 + (-0.894 - 0.446i)T \) |
| 31 | \( 1 + (-0.0275 + 0.999i)T \) |
| 37 | \( 1 + (0.0275 + 0.999i)T \) |
| 41 | \( 1 + (-0.677 + 0.735i)T \) |
| 43 | \( 1 + (-0.934 + 0.355i)T \) |
| 47 | \( 1 + (0.224 + 0.974i)T \) |
| 53 | \( 1 + (-0.874 - 0.485i)T \) |
| 59 | \( 1 + (-0.942 - 0.335i)T \) |
| 61 | \( 1 + (0.949 + 0.314i)T \) |
| 67 | \( 1 + (-0.739 - 0.673i)T \) |
| 71 | \( 1 + (-0.980 + 0.197i)T \) |
| 73 | \( 1 + (-0.996 + 0.0880i)T \) |
| 79 | \( 1 + (-0.834 - 0.551i)T \) |
| 83 | \( 1 + (0.828 + 0.560i)T \) |
| 89 | \( 1 + (-0.889 + 0.456i)T \) |
| 97 | \( 1 + (-0.991 + 0.131i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−18.2682931067639769767501082253, −17.50351711571190464937608054701, −16.96980601608482856994269343092, −16.54739072571136777563348497733, −15.5195931046839880535380258289, −14.81428054033424907369578489467, −14.49776289592603916568500282992, −13.65594818352383093772359288993, −12.95165508487739296633417008553, −11.9282834541517987881112722450, −11.19568481734805670338483319815, −10.46809112448176608576702988969, −9.85373294316805816667745915260, −9.14531113877571035578599876499, −8.85596800485550453383521175221, −7.55683372271910685764723436187, −6.99951736156511076776693402447, −6.6150816221981836309773030834, −5.66084384365091963165772069017, −4.986728604699639046201099797233, −4.19407858952779927387111375526, −2.76799288964956847092367918784, −2.08029028808790066270488767051, −1.51948859983589756301142196897, −0.064224098022180461135226534809,
1.237245834650804410701553280428, 1.6638637568244541220635504711, 2.80230062140498881661586251231, 3.29543837634357409164106863205, 4.35019201763738949974613706131, 5.27405460269677980856177049640, 6.1336890239241611617092234719, 6.69623449564315105924123882558, 7.85007763732793946497828445473, 8.39228236513350310576868690403, 8.96970908766710546239227245845, 9.73895660380874514363794633179, 10.38581506943154004974600067017, 10.907717851682496149292225130543, 11.69877837575812660248429172668, 12.64464877985987711867182502865, 13.02122447785520167481405108533, 13.61354724252875003596814329103, 14.682409199831826778182058935958, 15.32536393837941704671709178753, 16.33565761641425666046673077389, 16.87409051820584254743398367068, 17.35414617641392377097834579405, 17.88288236059270405596200804709, 18.82568936408742248023774902175