L(s) = 1 | + (−0.707 + 0.707i)2-s + (−0.707 + 0.707i)3-s − 1.00i·4-s + (−2.23 − 0.0168i)5-s − 1.00i·6-s + (−2.83 + 2.83i)7-s + (0.707 + 0.707i)8-s − 1.00i·9-s + (1.59 − 1.56i)10-s − 1.81i·11-s + (0.707 + 0.707i)12-s + (−3.73 + 3.73i)13-s − 4.01i·14-s + (1.59 − 1.56i)15-s − 1.00·16-s + (0.414 + 0.414i)17-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s + (−0.408 + 0.408i)3-s − 0.500i·4-s + (−0.999 − 0.00754i)5-s − 0.408i·6-s + (−1.07 + 1.07i)7-s + (0.250 + 0.250i)8-s − 0.333i·9-s + (0.503 − 0.496i)10-s − 0.546i·11-s + (0.204 + 0.204i)12-s + (−1.03 + 1.03i)13-s − 1.07i·14-s + (0.411 − 0.405i)15-s − 0.250·16-s + (0.100 + 0.100i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.294961 - 0.0769470i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.294961 - 0.0769470i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (2.23 + 0.0168i)T \) |
| 31 | \( 1 + (-0.282 - 5.56i)T \) |
good | 7 | \( 1 + (2.83 - 2.83i)T - 7iT^{2} \) |
| 11 | \( 1 + 1.81iT - 11T^{2} \) |
| 13 | \( 1 + (3.73 - 3.73i)T - 13iT^{2} \) |
| 17 | \( 1 + (-0.414 - 0.414i)T + 17iT^{2} \) |
| 19 | \( 1 + 5.16iT - 19T^{2} \) |
| 23 | \( 1 + (1.45 - 1.45i)T - 23iT^{2} \) |
| 29 | \( 1 - 1.79T + 29T^{2} \) |
| 37 | \( 1 + (7.51 + 7.51i)T + 37iT^{2} \) |
| 41 | \( 1 - 8.55T + 41T^{2} \) |
| 43 | \( 1 + (1.19 - 1.19i)T - 43iT^{2} \) |
| 47 | \( 1 + (-5.64 + 5.64i)T - 47iT^{2} \) |
| 53 | \( 1 + (-10.0 + 10.0i)T - 53iT^{2} \) |
| 59 | \( 1 + 11.6iT - 59T^{2} \) |
| 61 | \( 1 - 14.9iT - 61T^{2} \) |
| 67 | \( 1 + (-8.55 + 8.55i)T - 67iT^{2} \) |
| 71 | \( 1 + 8.79T + 71T^{2} \) |
| 73 | \( 1 + (8.90 - 8.90i)T - 73iT^{2} \) |
| 79 | \( 1 - 1.10T + 79T^{2} \) |
| 83 | \( 1 + (-4.92 + 4.92i)T - 83iT^{2} \) |
| 89 | \( 1 + 2.43T + 89T^{2} \) |
| 97 | \( 1 + (-2.65 + 2.65i)T - 97iT^{2} \) |
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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
−9.824338879929555175412971450946, −9.010013580507387881584103628973, −8.631148125698949311347443073108, −7.25348107738639591778354099350, −6.76018669552444007376458068231, −5.71538167711627698872509792124, −4.86770602965703367716536836718, −3.72066806733582847334937111425, −2.52898939998822675068008129959, −0.24612317414337087332373279673,
0.874102791881834745416196949765, 2.68261962364726726606294228717, 3.71242220433105427935181304106, 4.57740757157603432388029467730, 5.97613699748177152627687533595, 7.14268495916976016661933020204, 7.49650994500112235169060941740, 8.289195661492854041588039471173, 9.578170431185757186901747521172, 10.29318862566202623641480963149