L(s) = 1 | + (−0.707 + 0.707i)2-s + (−1.47 − 0.909i)3-s − 1.00i·4-s + (2.03 − 0.932i)5-s + (1.68 − 0.399i)6-s + (−2.61 − 2.61i)7-s + (0.707 + 0.707i)8-s + (1.34 + 2.68i)9-s + (−0.777 + 2.09i)10-s + 2.69i·11-s + (−0.909 + 1.47i)12-s + (−3.30 + 3.30i)13-s + 3.69·14-s + (−3.84 − 0.473i)15-s − 1.00·16-s + (−5.75 + 5.75i)17-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s + (−0.851 − 0.525i)3-s − 0.500i·4-s + (0.908 − 0.417i)5-s + (0.688 − 0.163i)6-s + (−0.987 − 0.987i)7-s + (0.250 + 0.250i)8-s + (0.448 + 0.893i)9-s + (−0.245 + 0.662i)10-s + 0.812i·11-s + (−0.262 + 0.425i)12-s + (−0.917 + 0.917i)13-s + 0.987·14-s + (−0.992 − 0.122i)15-s − 0.250·16-s + (−1.39 + 1.39i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.346 - 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.346 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.542371 + 0.377651i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.542371 + 0.377651i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.47 + 0.909i)T \) |
| 5 | \( 1 + (-2.03 + 0.932i)T \) |
| 31 | \( 1 - T \) |
good | 7 | \( 1 + (2.61 + 2.61i)T + 7iT^{2} \) |
| 11 | \( 1 - 2.69iT - 11T^{2} \) |
| 13 | \( 1 + (3.30 - 3.30i)T - 13iT^{2} \) |
| 17 | \( 1 + (5.75 - 5.75i)T - 17iT^{2} \) |
| 19 | \( 1 + 5.51iT - 19T^{2} \) |
| 23 | \( 1 + (-6.03 - 6.03i)T + 23iT^{2} \) |
| 29 | \( 1 - 5.99T + 29T^{2} \) |
| 37 | \( 1 + (-6.06 - 6.06i)T + 37iT^{2} \) |
| 41 | \( 1 - 2.10iT - 41T^{2} \) |
| 43 | \( 1 + (5.90 - 5.90i)T - 43iT^{2} \) |
| 47 | \( 1 + (-5.62 + 5.62i)T - 47iT^{2} \) |
| 53 | \( 1 + (1.97 + 1.97i)T + 53iT^{2} \) |
| 59 | \( 1 - 3.53T + 59T^{2} \) |
| 61 | \( 1 - 1.96T + 61T^{2} \) |
| 67 | \( 1 + (3.81 + 3.81i)T + 67iT^{2} \) |
| 71 | \( 1 - 9.25iT - 71T^{2} \) |
| 73 | \( 1 + (1.58 - 1.58i)T - 73iT^{2} \) |
| 79 | \( 1 - 5.86iT - 79T^{2} \) |
| 83 | \( 1 + (-2.85 - 2.85i)T + 83iT^{2} \) |
| 89 | \( 1 + 5.13T + 89T^{2} \) |
| 97 | \( 1 + (-1.82 - 1.82i)T + 97iT^{2} \) |
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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
−10.02489861429153077432403125061, −9.580054483995906438236868438129, −8.625164720778049163939237100212, −7.33106011771254286977635479174, −6.68886132263054225889893766971, −6.39548527095445532242280411113, −5.02058553151789973902135141099, −4.43780706426630422289371242947, −2.34923653116481688391229050879, −1.14449817584577029085106891632,
0.46155395896109859447216711577, 2.52666956057406166933922880786, 3.10881557748257150082110709497, 4.70566286451425478699712169671, 5.65947083638634431304360261084, 6.34974443189170044934157756844, 7.16174778245139721525493741114, 8.720692393260876078343794815771, 9.258370815005574265220733188539, 10.00517052364209963100854704334