L(s) = 1 | + (5.19 + 0.160i)3-s + 7.59i·5-s − 7i·7-s + (26.9 + 1.66i)9-s − 0.0298·11-s + 40.1·13-s + (−1.21 + 39.4i)15-s + 0.439i·17-s + 0.434i·19-s + (1.12 − 36.3i)21-s + 139.·23-s + 67.2·25-s + (139. + 12.9i)27-s + 273. i·29-s + 149. i·31-s + ⋯ |
L(s) = 1 | + (0.999 + 0.0308i)3-s + 0.679i·5-s − 0.377i·7-s + (0.998 + 0.0617i)9-s − 0.000818·11-s + 0.856·13-s + (−0.0209 + 0.679i)15-s + 0.00627i·17-s + 0.00524i·19-s + (0.0116 − 0.377i)21-s + 1.26·23-s + 0.538·25-s + (0.995 + 0.0925i)27-s + 1.75i·29-s + 0.866i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.473i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.881 - 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.874445732\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.874445732\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-5.19 - 0.160i)T \) |
| 7 | \( 1 + 7iT \) |
good | 5 | \( 1 - 7.59iT - 125T^{2} \) |
| 11 | \( 1 + 0.0298T + 1.33e3T^{2} \) |
| 13 | \( 1 - 40.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 0.439iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 0.434iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 139.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 273. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 149. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 290.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 57.4iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 432. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 317.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 147. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 488.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 441.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 440. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 202.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 542.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 325. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 376.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 496. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 373.T + 9.12e5T^{2} \) |
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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
−10.75969452080749356030038691401, −10.51609017303099873726878376015, −9.111078174475128081338092760409, −8.556454343341476557458394806211, −7.25474692901240231114919666271, −6.75478987903249055376734591459, −5.11426549327210516634761990206, −3.72357430967395266453664195070, −2.93565934140980849110267095809, −1.38745644011304726388951867731,
1.10992470486021770533628338529, 2.52805233079852465880482043205, 3.79806215776009413566042522084, 4.90415181462017761285452189436, 6.21799625564752589447872470775, 7.44308075855613370829106922103, 8.443346262384909358007073335422, 8.995757110022897164979212731755, 9.880629840977085931410390037296, 11.03660305864676354085481372908