| L(s) = 1 | + (−0.884 − 1.10i)2-s + (0.846 − 1.51i)3-s + (−0.436 + 1.95i)4-s + (0.448 + 2.19i)5-s + (−2.41 + 0.401i)6-s + 2.41i·7-s + (2.53 − 1.24i)8-s + (−1.56 − 2.55i)9-s + (2.02 − 2.43i)10-s + (4.39 − 3.19i)11-s + (2.57 + 2.31i)12-s + (3.38 + 2.45i)13-s + (2.66 − 2.13i)14-s + (3.68 + 1.17i)15-s + (−3.61 − 1.70i)16-s + (−3.42 + 1.11i)17-s + ⋯ |
| L(s) = 1 | + (−0.625 − 0.780i)2-s + (0.488 − 0.872i)3-s + (−0.218 + 0.975i)4-s + (0.200 + 0.979i)5-s + (−0.986 + 0.163i)6-s + 0.913i·7-s + (0.898 − 0.439i)8-s + (−0.522 − 0.852i)9-s + (0.639 − 0.769i)10-s + (1.32 − 0.963i)11-s + (0.744 + 0.667i)12-s + (0.937 + 0.681i)13-s + (0.713 − 0.571i)14-s + (0.952 + 0.303i)15-s + (−0.904 − 0.425i)16-s + (−0.830 + 0.269i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.675 + 0.737i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.675 + 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.10120 - 0.484896i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.10120 - 0.484896i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.884 + 1.10i)T \) |
| 3 | \( 1 + (-0.846 + 1.51i)T \) |
| 5 | \( 1 + (-0.448 - 2.19i)T \) |
| good | 7 | \( 1 - 2.41iT - 7T^{2} \) |
| 11 | \( 1 + (-4.39 + 3.19i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-3.38 - 2.45i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (3.42 - 1.11i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-7.02 + 2.28i)T + (15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-0.603 + 0.438i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (3.66 + 1.19i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (0.927 - 0.301i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-4.42 - 3.21i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-1.05 + 1.45i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 4.88iT - 43T^{2} \) |
| 47 | \( 1 + (2.13 - 6.58i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (4.55 + 1.48i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (7.42 + 5.39i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (8.05 - 5.85i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (1.52 - 0.495i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (-3.21 + 9.88i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-3.45 + 2.51i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (8.43 + 2.74i)T + (63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-2.68 - 8.27i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-7.10 - 9.77i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-0.0838 + 0.258i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−11.46833740019037656971933306199, −11.06754087546502586537862185477, −9.282964250236607408572558308786, −9.116998336416760417842970316905, −7.964578529669388165326474538703, −6.83021885323577933758052869456, −6.07148570315052715939905294648, −3.72300917123808859102417326162, −2.81537950829582909187249424604, −1.52834802026891009830792684285,
1.37720688305810955162647576870, 3.88590406964479247109391013963, 4.76871246725167607118651662811, 5.88342871664083358588621356475, 7.26807413187565091160326141455, 8.146193481558761681780418755310, 9.259442779823572848353050816740, 9.533011975102847333501864985879, 10.58478687631922736430770186700, 11.58065665438966859296947421690
