L(s) = 1 | + 0.672i·2-s − 2.65i·3-s + 1.54·4-s − 3.54i·5-s + 1.78·6-s + (−1.10 − 2.40i)7-s + 2.38i·8-s − 4.03·9-s + 2.38·10-s − 4.10i·12-s − 1.10·13-s + (1.61 − 0.742i)14-s − 9.41·15-s + 1.48·16-s + 4.17·17-s − 2.71i·18-s + ⋯ |
L(s) = 1 | + 0.475i·2-s − 1.53i·3-s + 0.773·4-s − 1.58i·5-s + 0.728·6-s + (−0.416 − 0.908i)7-s + 0.843i·8-s − 1.34·9-s + 0.755·10-s − 1.18i·12-s − 0.305·13-s + (0.432 − 0.198i)14-s − 2.43·15-s + 0.372·16-s + 1.01·17-s − 0.640i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.688 + 0.725i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.688 + 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.693162 - 1.61259i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.693162 - 1.61259i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (1.10 + 2.40i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 0.672iT - 2T^{2} \) |
| 3 | \( 1 + 2.65iT - 3T^{2} \) |
| 5 | \( 1 + 3.54iT - 5T^{2} \) |
| 13 | \( 1 + 1.10T + 13T^{2} \) |
| 17 | \( 1 - 4.17T + 17T^{2} \) |
| 19 | \( 1 - 1.24T + 19T^{2} \) |
| 23 | \( 1 - 2.85T + 23T^{2} \) |
| 29 | \( 1 - 2.04iT - 29T^{2} \) |
| 31 | \( 1 - 2.99iT - 31T^{2} \) |
| 37 | \( 1 - 2.76T + 37T^{2} \) |
| 41 | \( 1 + 5.49T + 41T^{2} \) |
| 43 | \( 1 + 1.73iT - 43T^{2} \) |
| 47 | \( 1 - 6.82iT - 47T^{2} \) |
| 53 | \( 1 - 1.07T + 53T^{2} \) |
| 59 | \( 1 - 2.28iT - 59T^{2} \) |
| 61 | \( 1 + 13.3T + 61T^{2} \) |
| 67 | \( 1 - 0.489T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 - 12.5T + 73T^{2} \) |
| 79 | \( 1 + 12.4iT - 79T^{2} \) |
| 83 | \( 1 - 1.61T + 83T^{2} \) |
| 89 | \( 1 + 2.30iT - 89T^{2} \) |
| 97 | \( 1 + 12.6iT - 97T^{2} \) |
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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
−9.740033145195673482575723341724, −8.683523100427502412476461074301, −7.85332843007010694324378608335, −7.40005244910073987729486040090, −6.57539428830440257679654455061, −5.71224874635027473075265908549, −4.78981574328594458455202321059, −3.16936003017839812973669191367, −1.68595494013924636971875541432, −0.871873922124263795320492388441,
2.37306581843072739914840158256, 3.14744969750225067240330088811, 3.70410880819395116963389544809, 5.21617947000307781732303022384, 6.11625825130959145035291597827, 6.92726728986398689589117485684, 7.964704675853364334904247682956, 9.366975206364195034031836822670, 9.875741710805956256860265297977, 10.45767884009112325146568449568