L(s) = 1 | + (0.382 + 0.923i)3-s + (−0.923 − 0.382i)5-s + (0.707 − 0.707i)7-s + (−0.707 + 0.707i)9-s + (−0.541 − 0.541i)13-s − i·15-s − 1.84i·19-s + (0.923 + 0.382i)21-s + (1 − i)23-s + (0.707 + 0.707i)25-s + (−0.923 − 0.382i)27-s + (−0.923 + 0.382i)35-s + (0.292 − 0.707i)39-s + (0.923 − 0.382i)45-s − 1.00i·49-s + ⋯ |
L(s) = 1 | + (0.382 + 0.923i)3-s + (−0.923 − 0.382i)5-s + (0.707 − 0.707i)7-s + (−0.707 + 0.707i)9-s + (−0.541 − 0.541i)13-s − i·15-s − 1.84i·19-s + (0.923 + 0.382i)21-s + (1 − i)23-s + (0.707 + 0.707i)25-s + (−0.923 − 0.382i)27-s + (−0.923 + 0.382i)35-s + (0.292 − 0.707i)39-s + (0.923 − 0.382i)45-s − 1.00i·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.154459525\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.154459525\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.382 - 0.923i)T \) |
| 5 | \( 1 + (0.923 + 0.382i)T \) |
| 7 | \( 1 + (-0.707 + 0.707i)T \) |
good | 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (0.541 + 0.541i)T + iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + 1.84iT - T^{2} \) |
| 23 | \( 1 + (-1 + i)T - iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 - 1.84T + T^{2} \) |
| 61 | \( 1 + 0.765T + T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 + 1.41iT - T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 - 1.41iT - T^{2} \) |
| 83 | \( 1 + (-1.30 + 1.30i)T - iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + iT^{2} \) |
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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
−8.766337277396928769802671986319, −8.062022227407298838175041595353, −7.43664674732840630658972111644, −6.70319470400455643415034433900, −5.20241506143048614740122080604, −4.83894813273920517120413078318, −4.20614666696370018470202407295, −3.29795402103987348238764876364, −2.44111770790349918237505979927, −0.70419419749827836186568341279,
1.37415554644104943027881642219, 2.30631115694318573261645016259, 3.24940346485489654429837971108, 4.05208492218932404416221390740, 5.15950855214477065241005726149, 5.93371577913785198420169899792, 6.84211449547595388459120570967, 7.49068191283867264546759958396, 8.028289322395749786300008215537, 8.629087272652292175704265081912