| L(s) = 1 | − i·3-s − 3.46i·5-s − 1.73·7-s + 2·9-s − 5.19i·13-s − 3.46·15-s − 3·17-s + i·19-s + 1.73i·21-s + 1.73·23-s − 6.99·25-s − 5i·27-s − 1.73i·29-s + 3.46·31-s + 5.99i·35-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 1.54i·5-s − 0.654·7-s + 0.666·9-s − 1.44i·13-s − 0.894·15-s − 0.727·17-s + 0.229i·19-s + 0.377i·21-s + 0.361·23-s − 1.39·25-s − 0.962i·27-s − 0.321i·29-s + 0.622·31-s + 1.01i·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.219309721\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.219309721\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - iT \) |
| good | 3 | \( 1 + iT - 3T^{2} \) |
| 5 | \( 1 + 3.46iT - 5T^{2} \) |
| 7 | \( 1 + 1.73T + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 5.19iT - 13T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 23 | \( 1 - 1.73T + 23T^{2} \) |
| 29 | \( 1 + 1.73iT - 29T^{2} \) |
| 31 | \( 1 - 3.46T + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 10iT - 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 - 5.19iT - 53T^{2} \) |
| 59 | \( 1 + 9iT - 59T^{2} \) |
| 61 | \( 1 + 10.3iT - 61T^{2} \) |
| 67 | \( 1 - 13iT - 67T^{2} \) |
| 71 | \( 1 - 3.46T + 71T^{2} \) |
| 73 | \( 1 + T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 18T + 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−9.464748671441932409039733944565, −8.305242105736197538967546392191, −7.993669900200742645136310306120, −6.84619942592678888942919242058, −6.01880978994877518340702791784, −5.04860448240018613682534738912, −4.29510467888817761876833666905, −3.01868337391652168154813455937, −1.57573625160565060131243997498, −0.52327539071794820116424229984,
1.99570936840885491694436235401, 3.10761673108875031430359833063, 3.92553215443835919497526020369, 4.81557542890447627150786534131, 6.22929062557113963510343636474, 6.83927656353014488929483641243, 7.27244607404216564447681596460, 8.677923727172064905131000703439, 9.513065209783156587392504017743, 10.08169268783341038303218263877
