L(s) = 1 | − i·2-s − 4-s + (−2.07 + 0.826i)5-s + 4.67i·7-s + i·8-s + (0.826 + 2.07i)10-s − 0.0451·11-s − 5.26i·13-s + 4.67·14-s + 16-s − 3.60i·17-s + 6.22·19-s + (2.07 − 0.826i)20-s + 0.0451i·22-s − 2.20i·23-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.5·4-s + (−0.929 + 0.369i)5-s + 1.76i·7-s + 0.353i·8-s + (0.261 + 0.656i)10-s − 0.0136·11-s − 1.46i·13-s + 1.24·14-s + 0.250·16-s − 0.875i·17-s + 1.42·19-s + (0.464 − 0.184i)20-s + 0.00962i·22-s − 0.459i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.369i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.929 - 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.242385349\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.242385349\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.07 - 0.826i)T \) |
| 37 | \( 1 - iT \) |
good | 7 | \( 1 - 4.67iT - 7T^{2} \) |
| 11 | \( 1 + 0.0451T + 11T^{2} \) |
| 13 | \( 1 + 5.26iT - 13T^{2} \) |
| 17 | \( 1 + 3.60iT - 17T^{2} \) |
| 19 | \( 1 - 6.22T + 19T^{2} \) |
| 23 | \( 1 + 2.20iT - 23T^{2} \) |
| 29 | \( 1 + 4.20T + 29T^{2} \) |
| 31 | \( 1 + 3.01T + 31T^{2} \) |
| 41 | \( 1 - 7.38T + 41T^{2} \) |
| 43 | \( 1 - 5.54iT - 43T^{2} \) |
| 47 | \( 1 - 4.28iT - 47T^{2} \) |
| 53 | \( 1 - 6.10iT - 53T^{2} \) |
| 59 | \( 1 - 13.5T + 59T^{2} \) |
| 61 | \( 1 + 4.27T + 61T^{2} \) |
| 67 | \( 1 - 12.3iT - 67T^{2} \) |
| 71 | \( 1 - 4.49T + 71T^{2} \) |
| 73 | \( 1 + 7.03iT - 73T^{2} \) |
| 79 | \( 1 - 8.72T + 79T^{2} \) |
| 83 | \( 1 + 0.880iT - 83T^{2} \) |
| 89 | \( 1 - 9.97T + 89T^{2} \) |
| 97 | \( 1 - 0.240iT - 97T^{2} \) |
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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
−8.841400960309479982937635207151, −7.87742708157636472854583974888, −7.55078728251489641186206074181, −6.26494207915933409499450137949, −5.43043911396622875897828813254, −4.95614572021263443641393898013, −3.70414797217783657629331417126, −2.91429992809403701656967970517, −2.47783655816558234561235745211, −0.859819334857564487901276513036,
0.53940026411688828613713336718, 1.62344997431544734599408318251, 3.63398472270167328905437470923, 3.84865184716299926590080838945, 4.65803506329269988787531628205, 5.49789391307077471892364960446, 6.64988435139270175694502076849, 7.22041709880022152244581660733, 7.61794498378735648076081707490, 8.379327814305137717946749053790