| L(s) = 1 | + i·2-s + (−1.19 + 1.25i)3-s − 4-s + (−1.25 − 1.19i)6-s + 3.22·7-s − i·8-s + (−0.166 − 2.99i)9-s + (−2.73 + 1.87i)11-s + (1.19 − 1.25i)12-s − 0.712·13-s + 3.22i·14-s + 16-s − 6.12i·17-s + (2.99 − 0.166i)18-s + 2.33i·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + (−0.687 + 0.726i)3-s − 0.5·4-s + (−0.513 − 0.485i)6-s + 1.22·7-s − 0.353i·8-s + (−0.0553 − 0.998i)9-s + (−0.825 + 0.564i)11-s + (0.343 − 0.363i)12-s − 0.197·13-s + 0.863i·14-s + 0.250·16-s − 1.48i·17-s + (0.706 − 0.0391i)18-s + 0.535i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.301 + 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.301 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2707661033\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2707661033\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (1.19 - 1.25i)T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + (2.73 - 1.87i)T \) |
| good | 7 | \( 1 - 3.22T + 7T^{2} \) |
| 13 | \( 1 + 0.712T + 13T^{2} \) |
| 17 | \( 1 + 6.12iT - 17T^{2} \) |
| 19 | \( 1 - 2.33iT - 19T^{2} \) |
| 23 | \( 1 + 3.03T + 23T^{2} \) |
| 29 | \( 1 + 5.27T + 29T^{2} \) |
| 31 | \( 1 + 9.55T + 31T^{2} \) |
| 37 | \( 1 + 9.73iT - 37T^{2} \) |
| 41 | \( 1 + 0.761T + 41T^{2} \) |
| 43 | \( 1 + 5.79T + 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 - 1.09T + 53T^{2} \) |
| 59 | \( 1 + 1.95iT - 59T^{2} \) |
| 61 | \( 1 - 2.19iT - 61T^{2} \) |
| 67 | \( 1 + 2.80iT - 67T^{2} \) |
| 71 | \( 1 + 11.7iT - 71T^{2} \) |
| 73 | \( 1 - 14.6T + 73T^{2} \) |
| 79 | \( 1 - 13.5iT - 79T^{2} \) |
| 83 | \( 1 - 7.72iT - 83T^{2} \) |
| 89 | \( 1 + 3.04iT - 89T^{2} \) |
| 97 | \( 1 + 10.6iT - 97T^{2} \) |
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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
−9.372963699960935772128770313296, −8.289939134156551248504529347934, −7.56529225305234743574090303817, −6.89663204143510699614260700059, −5.62577441329474642832480470786, −5.21247947852921162676690280222, −4.54491596588366807337084325539, −3.54127821712146655541123110183, −1.93860100992042170044775084674, −0.11176296115971510811646232244,
1.46770602117660731244688738115, 2.14371820124775663064411045472, 3.51236659256117861923816156421, 4.72620835670703281270892625630, 5.33564171558261210819089148215, 6.14839878404848420360886957002, 7.26293345127553383437454140008, 8.195290159429249569863942909682, 8.378333737523196450018296164879, 9.747277948244839450592630911608
