L(s) = 1 | + (1.25 + 2.53i)2-s + 3i·3-s + (−4.86 + 6.35i)4-s − 9.15i·5-s + (−7.60 + 3.75i)6-s + 27.4·7-s + (−22.2 − 4.36i)8-s − 9·9-s + (23.2 − 11.4i)10-s − 20.5i·11-s + (−19.0 − 14.5i)12-s − 32.0i·13-s + (34.3 + 69.5i)14-s + 27.4·15-s + (−16.7 − 61.7i)16-s − 111.·17-s + ⋯ |
L(s) = 1 | + (0.442 + 0.896i)2-s + 0.577i·3-s + (−0.607 + 0.794i)4-s − 0.818i·5-s + (−0.517 + 0.255i)6-s + 1.48·7-s + (−0.981 − 0.192i)8-s − 0.333·9-s + (0.734 − 0.362i)10-s − 0.562i·11-s + (−0.458 − 0.350i)12-s − 0.683i·13-s + (0.655 + 1.32i)14-s + 0.472·15-s + (−0.261 − 0.965i)16-s − 1.59·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.192 - 0.981i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.192 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.05857 + 0.870807i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.05857 + 0.870807i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.25 - 2.53i)T \) |
| 3 | \( 1 - 3iT \) |
good | 5 | \( 1 + 9.15iT - 125T^{2} \) |
| 7 | \( 1 - 27.4T + 343T^{2} \) |
| 11 | \( 1 + 20.5iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 32.0iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 111.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 129. iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 9.16T + 1.21e4T^{2} \) |
| 29 | \( 1 + 41.0iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 187.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 114. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 282.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 89.3iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 54.6T + 1.03e5T^{2} \) |
| 53 | \( 1 - 726. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 216. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 754. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 379. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 302.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 504.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 301.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 599. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 277.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 765.T + 9.12e5T^{2} \) |
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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
−17.21357036204838171694064133849, −16.20885536902406643947887304763, −15.05179427976862094357557660672, −14.03849563937045367865012755845, −12.61623161062397753039932514790, −11.07528357697740631256886944908, −8.923402948061377104839545913169, −7.957427606768841897834892410294, −5.59776002128949147467911890689, −4.35143178292738007343162123873,
2.16303243989996515704103791579, 4.72169953520010168620514549605, 6.91322891317605065833849060303, 8.910819404006661901345732694658, 10.92415371421557910430604356644, 11.51497629264453074409614985873, 13.12798322866584959052141297608, 14.29404723064282305545418114153, 15.12554879465913182870736894004, 17.70141927153466442496081440605