| L(s) = 1 | + (−0.00461 − 0.0117i)2-s + (−0.562 − 7.50i)3-s + (5.86 − 5.44i)4-s + (−14.2 + 9.69i)5-s + (−0.0856 + 0.0412i)6-s + (−10.4 − 15.2i)7-s + (−0.182 − 0.0877i)8-s + (−29.2 + 4.41i)9-s + (0.179 + 0.122i)10-s + (35.6 + 5.36i)11-s + (−44.1 − 40.9i)12-s + (45.3 − 56.9i)13-s + (−0.131 + 0.193i)14-s + (80.6 + 101. i)15-s + (4.78 − 63.8i)16-s + (3.02 − 0.934i)17-s + ⋯ |
| L(s) = 1 | + (−0.00163 − 0.00416i)2-s + (−0.108 − 1.44i)3-s + (0.733 − 0.680i)4-s + (−1.27 + 0.866i)5-s + (−0.00583 + 0.00280i)6-s + (−0.563 − 0.825i)7-s + (−0.00805 − 0.00387i)8-s + (−1.08 + 0.163i)9-s + (0.00568 + 0.00387i)10-s + (0.976 + 0.147i)11-s + (−1.06 − 0.984i)12-s + (0.968 − 1.21i)13-s + (−0.00251 + 0.00369i)14-s + (1.38 + 1.74i)15-s + (0.0747 − 0.997i)16-s + (0.0432 − 0.0133i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.480 + 0.877i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.480 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.601746 - 1.01560i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.601746 - 1.01560i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (10.4 + 15.2i)T \) |
| good | 2 | \( 1 + (0.00461 + 0.0117i)T + (-5.86 + 5.44i)T^{2} \) |
| 3 | \( 1 + (0.562 + 7.50i)T + (-26.6 + 4.02i)T^{2} \) |
| 5 | \( 1 + (14.2 - 9.69i)T + (45.6 - 116. i)T^{2} \) |
| 11 | \( 1 + (-35.6 - 5.36i)T + (1.27e3 + 392. i)T^{2} \) |
| 13 | \( 1 + (-45.3 + 56.9i)T + (-488. - 2.14e3i)T^{2} \) |
| 17 | \( 1 + (-3.02 + 0.934i)T + (4.05e3 - 2.76e3i)T^{2} \) |
| 19 | \( 1 + (44.3 - 76.7i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-145. - 44.8i)T + (1.00e4 + 6.85e3i)T^{2} \) |
| 29 | \( 1 + (-21.1 + 92.5i)T + (-2.19e4 - 1.05e4i)T^{2} \) |
| 31 | \( 1 + (-95.5 - 165. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-130. - 121. i)T + (3.78e3 + 5.05e4i)T^{2} \) |
| 41 | \( 1 + (258. + 124. i)T + (4.29e4 + 5.38e4i)T^{2} \) |
| 43 | \( 1 + (90.3 - 43.5i)T + (4.95e4 - 6.21e4i)T^{2} \) |
| 47 | \( 1 + (53.6 + 136. i)T + (-7.61e4 + 7.06e4i)T^{2} \) |
| 53 | \( 1 + (156. - 145. i)T + (1.11e4 - 1.48e5i)T^{2} \) |
| 59 | \( 1 + (-19.8 - 13.5i)T + (7.50e4 + 1.91e5i)T^{2} \) |
| 61 | \( 1 + (-179. - 166. i)T + (1.69e4 + 2.26e5i)T^{2} \) |
| 67 | \( 1 + (106. + 185. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-172. - 755. i)T + (-3.22e5 + 1.55e5i)T^{2} \) |
| 73 | \( 1 + (-310. + 790. i)T + (-2.85e5 - 2.64e5i)T^{2} \) |
| 79 | \( 1 + (-204. + 355. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-762. - 956. i)T + (-1.27e5 + 5.57e5i)T^{2} \) |
| 89 | \( 1 + (251. - 37.9i)T + (6.73e5 - 2.07e5i)T^{2} \) |
| 97 | \( 1 - 52.5T + 9.12e5T^{2} \) |
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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
−14.76876597178738808597755778241, −13.55237285311242464313345280670, −12.29395951878890228805301056005, −11.32932108237298327358442570660, −10.37974798621373049064744860015, −8.001221075812836138782904715736, −6.99712615799909171688319824363, −6.33503227418225567239297152619, −3.35320304377660974881137719154, −1.00230215129235707707879419030,
3.50607939578151755360257766926, 4.53257420595460411101310793226, 6.60276784653270638343383627070, 8.576152251245106390743313219778, 9.180596348402103138708627616816, 11.17577920929196662609121852215, 11.67714057066178451746890970964, 12.87144067185469765362380737128, 15.00025533880361466493490481742, 15.68559896951539155917949316575
