L(s) = 1 | + (0.391 + 1.46i)2-s + (1.50 − 0.852i)3-s + (−0.246 + 0.142i)4-s + (1.82 − 1.29i)5-s + (1.83 + 1.86i)6-s + (1.83 + 1.83i)8-s + (1.54 − 2.57i)9-s + (2.60 + 2.15i)10-s + (0.791 − 0.457i)11-s + (−0.250 + 0.425i)12-s + (−3.07 + 3.07i)13-s + (1.64 − 3.50i)15-s + (−2.24 + 3.88i)16-s + (1.16 + 0.311i)17-s + (4.35 + 1.24i)18-s + (−5.95 − 3.43i)19-s + ⋯ |
L(s) = 1 | + (0.276 + 1.03i)2-s + (0.870 − 0.492i)3-s + (−0.123 + 0.0712i)4-s + (0.815 − 0.578i)5-s + (0.749 + 0.762i)6-s + (0.648 + 0.648i)8-s + (0.514 − 0.857i)9-s + (0.822 + 0.682i)10-s + (0.238 − 0.137i)11-s + (−0.0723 + 0.122i)12-s + (−0.854 + 0.854i)13-s + (0.425 − 0.905i)15-s + (−0.561 + 0.971i)16-s + (0.281 + 0.0755i)17-s + (1.02 + 0.294i)18-s + (−1.36 − 0.788i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 - 0.489i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 - 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.81625 + 0.736008i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.81625 + 0.736008i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.50 + 0.852i)T \) |
| 5 | \( 1 + (-1.82 + 1.29i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.391 - 1.46i)T + (-1.73 + i)T^{2} \) |
| 11 | \( 1 + (-0.791 + 0.457i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (3.07 - 3.07i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1.16 - 0.311i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (5.95 + 3.43i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.88 + 0.505i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + 2.72T + 29T^{2} \) |
| 31 | \( 1 + (-2.31 - 4.01i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.774 - 0.207i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 - 0.922iT - 41T^{2} \) |
| 43 | \( 1 + (4.80 - 4.80i)T - 43iT^{2} \) |
| 47 | \( 1 + (-2.71 - 10.1i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-2.85 + 10.6i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (4.94 + 8.55i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.533 - 0.924i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.83 - 6.83i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 0.557iT - 71T^{2} \) |
| 73 | \( 1 + (2.10 + 0.564i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-2.62 - 1.51i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (2.38 + 2.38i)T + 83iT^{2} \) |
| 89 | \( 1 + (5.64 - 9.78i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.58 - 1.58i)T + 97iT^{2} \) |
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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
−10.18949305469239083765277387510, −9.264348224208204306545857266518, −8.632579540447798857519156438087, −7.78645465482956195481203244927, −6.74142103761034535054144314375, −6.39671439893748040845459164971, −5.12868203923727179060280730857, −4.30998993180283049795143992307, −2.57294235401324547573913968144, −1.60974171560472645900392041641,
1.82599349582666962098992727199, 2.60779633794103987048688256851, 3.47718800821249400024877889089, 4.46622466396570004293088001850, 5.66019163272534299371111285045, 6.96754572624880244000877414014, 7.73549282209814748077268036771, 8.886928367730366677039907745093, 9.840899094967451283650347072098, 10.31521889572585070525606294619