L(s) = 1 | + (0.309 − 0.951i)2-s + (−1.30 − 0.951i)3-s + (−0.809 − 0.587i)4-s + (−0.309 + 0.951i)5-s + (−1.30 + 0.951i)6-s + 7-s + (−0.809 + 0.587i)8-s + (0.500 + 1.53i)9-s + (0.809 + 0.587i)10-s + (0.499 + 1.53i)12-s + (0.5 + 1.53i)13-s + (0.309 − 0.951i)14-s + (1.30 − 0.951i)15-s + (0.309 + 0.951i)16-s + 1.61·18-s + (1.61 − 1.17i)19-s + ⋯ |
L(s) = 1 | + (0.309 − 0.951i)2-s + (−1.30 − 0.951i)3-s + (−0.809 − 0.587i)4-s + (−0.309 + 0.951i)5-s + (−1.30 + 0.951i)6-s + 7-s + (−0.809 + 0.587i)8-s + (0.500 + 1.53i)9-s + (0.809 + 0.587i)10-s + (0.499 + 1.53i)12-s + (0.5 + 1.53i)13-s + (0.309 − 0.951i)14-s + (1.30 − 0.951i)15-s + (0.309 + 0.951i)16-s + 1.61·18-s + (1.61 − 1.17i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7769386574\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7769386574\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.309 + 0.951i)T \) |
| 5 | \( 1 + (0.309 - 0.951i)T \) |
| 7 | \( 1 - T \) |
good | 3 | \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 11 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (-1.61 + 1.17i)T + (0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 29 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 71 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 97 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.891721024535960274384115057480, −9.016026076657478319656677813038, −7.75126551507613395142754451636, −7.12171918416976001221394753408, −6.25137562299408477533209363567, −5.42355156779689226948832990600, −4.63179191282320981710034938650, −3.53212327050473597218943166149, −2.14507414811926742479397129886, −1.25102279053126631379284794267,
0.856756275800495936277887822617, 3.49296635226964067475845602398, 4.35538968792354698104366005420, 5.06861614393463777531848420907, 5.54470601327486007760926593254, 6.18801518191244997682381618804, 7.62885541190765653042439619714, 8.111857166367127056275971952473, 8.948686116605942740699993239210, 9.953256462044768592643011694813