L(s) = 1 | + (17.9 − 17.9i)7-s + 4.24i·11-s + (17.9 + 17.9i)13-s + (76.0 + 76.0i)17-s − 26i·19-s + (−76.0 + 76.0i)23-s + 110.·29-s + 52·31-s + (17.9 − 17.9i)37-s + 199. i·41-s + (−304. − 304. i)47-s − 299i·49-s + (380. − 380. i)53-s + 717.·59-s + 350·61-s + ⋯ |
L(s) = 1 | + (0.967 − 0.967i)7-s + 0.116i·11-s + (0.382 + 0.382i)13-s + (1.08 + 1.08i)17-s − 0.313i·19-s + (−0.689 + 0.689i)23-s + 0.706·29-s + 0.301·31-s + (0.0796 − 0.0796i)37-s + 0.759i·41-s + (−0.943 − 0.943i)47-s − 0.871i·49-s + (0.985 − 0.985i)53-s + 1.58·59-s + 0.734·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0618i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.497379570\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.497379570\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-17.9 + 17.9i)T - 343iT^{2} \) |
| 11 | \( 1 - 4.24iT - 1.33e3T^{2} \) |
| 13 | \( 1 + (-17.9 - 17.9i)T + 2.19e3iT^{2} \) |
| 17 | \( 1 + (-76.0 - 76.0i)T + 4.91e3iT^{2} \) |
| 19 | \( 1 + 26iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (76.0 - 76.0i)T - 1.21e4iT^{2} \) |
| 29 | \( 1 - 110.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 52T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-17.9 + 17.9i)T - 5.06e4iT^{2} \) |
| 41 | \( 1 - 199. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 7.95e4iT^{2} \) |
| 47 | \( 1 + (304. + 304. i)T + 1.03e5iT^{2} \) |
| 53 | \( 1 + (-380. + 380. i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 - 717.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 350T + 2.26e5T^{2} \) |
| 67 | \( 1 + (465. - 465. i)T - 3.00e5iT^{2} \) |
| 71 | \( 1 - 517. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (465. + 465. i)T + 3.89e5iT^{2} \) |
| 79 | \( 1 + 1.00e3iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (456. - 456. i)T - 5.71e5iT^{2} \) |
| 89 | \( 1 - 1.42e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-788. + 788. i)T - 9.12e5iT^{2} \) |
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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
−10.01578151768177751584060465527, −8.687351797812056226497550575950, −8.031730683898188711770180936429, −7.28489167820799955799887141000, −6.29622205700100653781909358594, −5.26182454602167740971113481826, −4.28733512489869782055734614604, −3.49718391760351759030017753352, −1.89268615368323884625043736894, −0.941213901657129994184017913175,
0.873704699646567308194731687438, 2.16674387115314609821442424307, 3.18878118516480343338850575244, 4.51958345064001300097707000161, 5.38979888193575075046209349905, 6.08644195915708708449765247787, 7.32430365715835906472930855098, 8.182626356835119017064518627937, 8.720045294107330407125699126328, 9.759142004077747011753397024635