L(s) = 1 | + (0.244 − 1.39i)2-s + 1.68i·3-s + (−1.88 − 0.679i)4-s + 5-s + (2.34 + 0.411i)6-s + (0.695 − 2.55i)7-s + (−1.40 + 2.45i)8-s + 0.163·9-s + (0.244 − 1.39i)10-s + 1.45·11-s + (1.14 − 3.16i)12-s + 5.12·13-s + (−3.38 − 1.59i)14-s + 1.68i·15-s + (3.07 + 2.55i)16-s − 0.313i·17-s + ⋯ |
L(s) = 1 | + (0.172 − 0.984i)2-s + 0.972i·3-s + (−0.940 − 0.339i)4-s + 0.447·5-s + (0.957 + 0.167i)6-s + (0.262 − 0.964i)7-s + (−0.497 + 0.867i)8-s + 0.0544·9-s + (0.0771 − 0.440i)10-s + 0.438·11-s + (0.330 − 0.914i)12-s + 1.42·13-s + (−0.904 − 0.425i)14-s + 0.434i·15-s + (0.768 + 0.639i)16-s − 0.0761i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.706 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.706 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.36041 - 0.564342i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.36041 - 0.564342i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.244 + 1.39i)T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (-0.695 + 2.55i)T \) |
good | 3 | \( 1 - 1.68iT - 3T^{2} \) |
| 11 | \( 1 - 1.45T + 11T^{2} \) |
| 13 | \( 1 - 5.12T + 13T^{2} \) |
| 17 | \( 1 + 0.313iT - 17T^{2} \) |
| 19 | \( 1 + 0.250iT - 19T^{2} \) |
| 23 | \( 1 + 4.27iT - 23T^{2} \) |
| 29 | \( 1 - 1.63iT - 29T^{2} \) |
| 31 | \( 1 + 8.96T + 31T^{2} \) |
| 37 | \( 1 + 3.47iT - 37T^{2} \) |
| 41 | \( 1 - 9.88iT - 41T^{2} \) |
| 43 | \( 1 + 8.65T + 43T^{2} \) |
| 47 | \( 1 + 7.77T + 47T^{2} \) |
| 53 | \( 1 + 1.90iT - 53T^{2} \) |
| 59 | \( 1 - 7.73iT - 59T^{2} \) |
| 61 | \( 1 - 0.415T + 61T^{2} \) |
| 67 | \( 1 - 15.2T + 67T^{2} \) |
| 71 | \( 1 - 1.50iT - 71T^{2} \) |
| 73 | \( 1 + 10.7iT - 73T^{2} \) |
| 79 | \( 1 - 9.36iT - 79T^{2} \) |
| 83 | \( 1 - 3.45iT - 83T^{2} \) |
| 89 | \( 1 - 9.12iT - 89T^{2} \) |
| 97 | \( 1 + 16.5iT - 97T^{2} \) |
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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
−11.31101395679730223713316976804, −10.83578571077486631148912025356, −10.03944663990203768011615753328, −9.266131952749576177193492782144, −8.291196415177245539609758732828, −6.62135795505235130166193644835, −5.23450287584152511579364857435, −4.20652402429796836084739317786, −3.44823612157328506540109045523, −1.45598503723112908233610765494,
1.65985738756740725443342994207, 3.65520294426163686566030652052, 5.28718464778991563038783777154, 6.13664936212903125801363674864, 6.91478489501610852665081051040, 8.047401136288229093867503711828, 8.797362169568288240303233828419, 9.730475158977357273857756964253, 11.28907582669758388413042936676, 12.29624312549755722173669294492