L(s) = 1 | + (−0.550 − 1.69i)2-s + (0.809 + 0.587i)3-s + (−1.76 + 1.27i)4-s + (0.550 − 1.69i)6-s + (1.69 + 1.23i)8-s + (0.309 + 0.951i)9-s + (−0.987 − 0.156i)11-s − 2.17·12-s + (0.587 + 1.80i)13-s + (0.481 − 1.48i)16-s + (−0.610 + 1.87i)17-s + (1.44 − 1.04i)18-s + (−0.951 − 0.690i)19-s + (0.278 + 1.76i)22-s + 0.907·23-s + (0.647 + 1.99i)24-s + ⋯ |
L(s) = 1 | + (−0.550 − 1.69i)2-s + (0.809 + 0.587i)3-s + (−1.76 + 1.27i)4-s + (0.550 − 1.69i)6-s + (1.69 + 1.23i)8-s + (0.309 + 0.951i)9-s + (−0.987 − 0.156i)11-s − 2.17·12-s + (0.587 + 1.80i)13-s + (0.481 − 1.48i)16-s + (−0.610 + 1.87i)17-s + (1.44 − 1.04i)18-s + (−0.951 − 0.690i)19-s + (0.278 + 1.76i)22-s + 0.907·23-s + (0.647 + 1.99i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8131426719\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8131426719\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.809 - 0.587i)T \) |
| 11 | \( 1 + (0.987 + 0.156i)T \) |
| 61 | \( 1 + (0.309 - 0.951i)T \) |
good | 2 | \( 1 + (0.550 + 1.69i)T + (-0.809 + 0.587i)T^{2} \) |
| 5 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 7 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (-0.587 - 1.80i)T + (-0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (0.610 - 1.87i)T + (-0.809 - 0.587i)T^{2} \) |
| 19 | \( 1 + (0.951 + 0.690i)T + (0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 - 0.907T + T^{2} \) |
| 29 | \( 1 + (0.734 - 0.533i)T + (0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.550 - 1.69i)T + (-0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (0.253 - 0.183i)T + (0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (-0.437 + 1.34i)T + (-0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 79 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 89 | \( 1 - 1.97T + T^{2} \) |
| 97 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.336929154832688404564314597204, −8.751820783514707359801544574318, −8.467903935721223449065730720121, −7.39311678535344379720476929835, −6.15954234466438213923559779687, −4.66662071898105129600216741018, −4.14103014724390735943653857330, −3.40005783515136628396722255583, −2.27177855063693084980982456094, −1.79171537658583323397369183110,
0.62717365881658090889106592400, 2.40853350610774955147545431173, 3.53107258319266281330594764090, 4.88996722618729370298071563825, 5.57840257805418243254082078867, 6.38245418203761754687778292771, 7.25774315447076356739393037218, 7.75475622602943396681157217545, 8.242114833971634052657487660891, 9.009320017199039480735817569662