L(s) = 1 | + (2.45 − 1.41i)2-s + (4.02 − 6.91i)4-s + 16.9i·5-s + 7i·7-s + (0.100 − 22.6i)8-s + (23.8 + 41.4i)10-s − 13.3·11-s + 76.5·13-s + (9.87 + 17.1i)14-s + (−31.6 − 55.6i)16-s + 99.7i·17-s + 121. i·19-s + (117. + 68.0i)20-s + (−32.7 + 18.8i)22-s + 151.·23-s + ⋯ |
L(s) = 1 | + (0.866 − 0.498i)2-s + (0.502 − 0.864i)4-s + 1.51i·5-s + 0.377i·7-s + (0.00445 − 0.999i)8-s + (0.754 + 1.31i)10-s − 0.366·11-s + 1.63·13-s + (0.188 + 0.327i)14-s + (−0.494 − 0.868i)16-s + 1.42i·17-s + 1.46i·19-s + (1.30 + 0.760i)20-s + (−0.317 + 0.182i)22-s + 1.37·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.415i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.909 - 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(3.037944280\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.037944280\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-2.45 + 1.41i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 7iT \) |
good | 5 | \( 1 - 16.9iT - 125T^{2} \) |
| 11 | \( 1 + 13.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 76.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 99.7iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 121. iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 151.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 197. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 268. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 19.2T + 5.06e4T^{2} \) |
| 41 | \( 1 + 129. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 490. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 160.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 126. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 285.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 467.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 323. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 673.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 338.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 869. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 532.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 140. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 652.T + 9.12e5T^{2} \) |
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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
−11.56526275818581494285091291255, −10.62299023104127186815344210104, −10.46170020472003302699212965280, −8.852953368704579867478496748667, −7.44527529629927210037559275461, −6.28130125830159478769417251989, −5.73444351906465791004204968939, −3.89785790606092090683870903112, −3.13527339043049981109433382845, −1.73891157986315579112247832405,
0.997995566748957443587308279678, 3.04495402556281078567470048472, 4.53616644650326031747530448482, 5.08261949086004010696289637493, 6.34592178769509275327731524817, 7.49881769383431576303216516521, 8.576789612259652707504250080242, 9.226901248647620965959380906412, 10.97292327575493332668357181325, 11.67651637176744157607155642632