L(s) = 1 | − 1.08·3-s − 3.34i·5-s − i·7-s − 1.81·9-s + i·11-s + (−3.52 − 0.743i)13-s + 3.64i·15-s + 4.31·17-s + 7.13i·19-s + 1.08i·21-s + 1.49·23-s − 6.22·25-s + 5.23·27-s + 2.80·29-s + 6.91i·31-s + ⋯ |
L(s) = 1 | − 0.627·3-s − 1.49i·5-s − 0.377i·7-s − 0.605·9-s + 0.301i·11-s + (−0.978 − 0.206i)13-s + 0.940i·15-s + 1.04·17-s + 1.63i·19-s + 0.237i·21-s + 0.310·23-s − 1.24·25-s + 1.00·27-s + 0.521·29-s + 1.24i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.206i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 + 0.206i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.120912941\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.120912941\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 - iT \) |
| 13 | \( 1 + (3.52 + 0.743i)T \) |
good | 3 | \( 1 + 1.08T + 3T^{2} \) |
| 5 | \( 1 + 3.34iT - 5T^{2} \) |
| 17 | \( 1 - 4.31T + 17T^{2} \) |
| 19 | \( 1 - 7.13iT - 19T^{2} \) |
| 23 | \( 1 - 1.49T + 23T^{2} \) |
| 29 | \( 1 - 2.80T + 29T^{2} \) |
| 31 | \( 1 - 6.91iT - 31T^{2} \) |
| 37 | \( 1 + 3.56iT - 37T^{2} \) |
| 41 | \( 1 - 8.88iT - 41T^{2} \) |
| 43 | \( 1 + 6.06T + 43T^{2} \) |
| 47 | \( 1 - 4.61iT - 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 + 9.89iT - 59T^{2} \) |
| 61 | \( 1 - 3.97T + 61T^{2} \) |
| 67 | \( 1 - 12.7iT - 67T^{2} \) |
| 71 | \( 1 - 6.59iT - 71T^{2} \) |
| 73 | \( 1 + 12.0iT - 73T^{2} \) |
| 79 | \( 1 - 9.43T + 79T^{2} \) |
| 83 | \( 1 + 0.795iT - 83T^{2} \) |
| 89 | \( 1 - 1.02iT - 89T^{2} \) |
| 97 | \( 1 + 12.9iT - 97T^{2} \) |
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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
−8.314984331964209047942735220171, −7.891752716285879028931759514020, −6.95411201699001304731388046872, −5.99868368593938969760479177704, −5.27667483252431675495893596035, −4.93984054407908137458447835867, −4.00093208732278177671928688897, −2.98376563086286443320626042515, −1.60424023969449084278281462679, −0.74995152367802405096194387057,
0.52263421017929306538365008878, 2.38302629352681937266527046694, 2.79945016934788766391007942140, 3.72079238313281581901300962292, 4.96287695718869850563464206618, 5.51853054192179413858080820619, 6.35736766574734731276387667297, 6.91817087906129695082110403585, 7.51649260688430815097898272625, 8.449539865812959098195087501963