L(s) = 1 | + (−0.104 + 0.994i)5-s + 1.82·7-s + (−0.809 − 0.587i)9-s + (−1.47 + 1.07i)11-s + (0.413 + 1.27i)17-s + (0.309 + 0.951i)19-s + (1.30 − 0.951i)23-s + (−0.978 − 0.207i)25-s + (−0.190 + 1.81i)35-s − 1.95·43-s + (0.669 − 0.743i)45-s + (−0.309 + 0.951i)47-s + 2.33·49-s + (−0.913 − 1.58i)55-s + (1.58 − 1.14i)61-s + ⋯ |
L(s) = 1 | + (−0.104 + 0.994i)5-s + 1.82·7-s + (−0.809 − 0.587i)9-s + (−1.47 + 1.07i)11-s + (0.413 + 1.27i)17-s + (0.309 + 0.951i)19-s + (1.30 − 0.951i)23-s + (−0.978 − 0.207i)25-s + (−0.190 + 1.81i)35-s − 1.95·43-s + (0.669 − 0.743i)45-s + (−0.309 + 0.951i)47-s + 2.33·49-s + (−0.913 − 1.58i)55-s + (1.58 − 1.14i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.387 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.387 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.179463880\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.179463880\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.104 - 0.994i)T \) |
| 19 | \( 1 + (-0.309 - 0.951i)T \) |
good | 3 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 7 | \( 1 - 1.82T + T^{2} \) |
| 11 | \( 1 + (1.47 - 1.07i)T + (0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 17 | \( 1 + (-0.413 - 1.27i)T + (-0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 29 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 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 | \( 1 + 1.95T + T^{2} \) |
| 47 | \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \) |
| 53 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (-1.58 + 1.14i)T + (0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 71 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \) |
| 79 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 89 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 97 | \( 1 + (0.809 + 0.587i)T^{2} \) |
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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
−9.710009290362181276316828413603, −8.345248230680674185320636775281, −8.130168454911555029880143827207, −7.34164734963892603866965823743, −6.41541814247999582704642625758, −5.41245964774032406956058614312, −4.81865561328176641689260153488, −3.67283033647532118699740048407, −2.62866972466538092760993707399, −1.71228496827695187931748051540,
0.912382448147095896730278444607, 2.25974538650321310691659273504, 3.24634210267969985332637367493, 4.81988173236302244613455357025, 5.19352408786289804953531505884, 5.49236733893881082552420806842, 7.23668846902493335347689624635, 7.86973935496987571771981849942, 8.475167097148167537148945496359, 8.899554874781983254248300279775