| L(s) = 1 | + 2.56i·2-s + 1.56i·3-s − 4.56·4-s − 4·6-s + i·7-s − 6.56i·8-s + 0.561·9-s − 1.56·11-s − 7.12i·12-s + 0.438i·13-s − 2.56·14-s + 7.68·16-s + 0.438i·17-s + 1.43i·18-s + 7.12·19-s + ⋯ |
| L(s) = 1 | + 1.81i·2-s + 0.901i·3-s − 2.28·4-s − 1.63·6-s + 0.377i·7-s − 2.31i·8-s + 0.187·9-s − 0.470·11-s − 2.05i·12-s + 0.121i·13-s − 0.684·14-s + 1.92·16-s + 0.106i·17-s + 0.339i·18-s + 1.63·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.227177 - 0.962340i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.227177 - 0.962340i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 2 | \( 1 - 2.56iT - 2T^{2} \) |
| 3 | \( 1 - 1.56iT - 3T^{2} \) |
| 11 | \( 1 + 1.56T + 11T^{2} \) |
| 13 | \( 1 - 0.438iT - 13T^{2} \) |
| 17 | \( 1 - 0.438iT - 17T^{2} \) |
| 19 | \( 1 - 7.12T + 19T^{2} \) |
| 23 | \( 1 - 3.12iT - 23T^{2} \) |
| 29 | \( 1 + 6.68T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 - 5.12T + 41T^{2} \) |
| 43 | \( 1 - 0.876iT - 43T^{2} \) |
| 47 | \( 1 - 8.68iT - 47T^{2} \) |
| 53 | \( 1 + 5.12iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 15.3T + 61T^{2} \) |
| 67 | \( 1 + 10.2iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 12.2iT - 73T^{2} \) |
| 79 | \( 1 - 2.43T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 - 1.12T + 89T^{2} \) |
| 97 | \( 1 + 5.80iT - 97T^{2} \) |
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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
−13.65561250647546665660607995928, −12.68691417409336477942535408235, −11.13263982060705219395093774094, −9.684308185155923627845325571145, −9.261338228670521472303535516259, −7.944618598253357275994836845905, −7.13642732478025094772087550949, −5.71978643248983676128468301785, −5.02314218023534529060404694834, −3.74986861588644646392379912603,
1.03512613646503030578168443128, 2.47410597584842334148276805115, 3.85566179114664969708336788933, 5.27504434173653871080047728415, 7.11259926744100988530058152192, 8.219272527875463299226035180402, 9.544419693925570063553528881529, 10.29160474035523459066367202414, 11.36020954057242596448877493838, 12.08763740568024894640142432218
