| L(s) = 1 | + 0.688·2-s − 1.52·4-s + 0.0401i·5-s + 0.246i·7-s − 2.42·8-s + 0.0276i·10-s − 2.30i·13-s + 0.169i·14-s + 1.38·16-s + 4.27·17-s + 6.20i·19-s − 0.0612i·20-s + 6.79i·23-s + 4.99·25-s − 1.58i·26-s + ⋯ |
| L(s) = 1 | + 0.486·2-s − 0.763·4-s + 0.0179i·5-s + 0.0932i·7-s − 0.858·8-s + 0.00872i·10-s − 0.638i·13-s + 0.0454i·14-s + 0.345·16-s + 1.03·17-s + 1.42i·19-s − 0.0136i·20-s + 1.41i·23-s + 0.999·25-s − 0.310i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.742 - 0.670i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.742 - 0.670i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.565938456\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.565938456\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.688T + 2T^{2} \) |
| 5 | \( 1 - 0.0401iT - 5T^{2} \) |
| 7 | \( 1 - 0.246iT - 7T^{2} \) |
| 13 | \( 1 + 2.30iT - 13T^{2} \) |
| 17 | \( 1 - 4.27T + 17T^{2} \) |
| 19 | \( 1 - 6.20iT - 19T^{2} \) |
| 23 | \( 1 - 6.79iT - 23T^{2} \) |
| 29 | \( 1 - 5.59T + 29T^{2} \) |
| 31 | \( 1 - 4.79T + 31T^{2} \) |
| 37 | \( 1 + 4.03T + 37T^{2} \) |
| 41 | \( 1 + 9.60T + 41T^{2} \) |
| 43 | \( 1 + 1.03iT - 43T^{2} \) |
| 47 | \( 1 - 11.1iT - 47T^{2} \) |
| 53 | \( 1 - 8.96iT - 53T^{2} \) |
| 59 | \( 1 - 2.78iT - 59T^{2} \) |
| 61 | \( 1 + 8.48iT - 61T^{2} \) |
| 67 | \( 1 - 7.94T + 67T^{2} \) |
| 71 | \( 1 - 3.32iT - 71T^{2} \) |
| 73 | \( 1 + 11.8iT - 73T^{2} \) |
| 79 | \( 1 + 3.01iT - 79T^{2} \) |
| 83 | \( 1 - 5.29T + 83T^{2} \) |
| 89 | \( 1 - 8.54iT - 89T^{2} \) |
| 97 | \( 1 + 3.02T + 97T^{2} \) |
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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
−10.00807525591840068378037295673, −9.166724690668300733168572853385, −8.264705616580002322997561481754, −7.65702125886717424269914926172, −6.34401523827955897108130108399, −5.54678942245472750420831982300, −4.86574963311835921316353900616, −3.70335066314544595744469995648, −3.02181105881152292032059965197, −1.18647733965341562787112141491,
0.74272296545633417627331677603, 2.60356432103051494671873697053, 3.62524864191272911987521553340, 4.69257762984014355829141528240, 5.16099028999520433533168656589, 6.41586876110430556109505642227, 7.05931219435177799349431547332, 8.544958076912415363211894419597, 8.654262868838531195168458638339, 9.863524277203738865369345728281
