| L(s) = 1 | − 2.58·3-s + i·5-s + (0.319 − 2.62i)7-s + 3.66·9-s − 0.0961i·11-s + 2.44i·13-s − 2.58i·15-s + 2.17i·17-s − 2.11·19-s + (−0.823 + 6.77i)21-s − 2.75i·23-s − 25-s − 1.70·27-s − 6.89·29-s − 2.00·31-s + ⋯ |
| L(s) = 1 | − 1.49·3-s + 0.447i·5-s + (0.120 − 0.992i)7-s + 1.22·9-s − 0.0290i·11-s + 0.676i·13-s − 0.666i·15-s + 0.527i·17-s − 0.484·19-s + (−0.179 + 1.47i)21-s − 0.573i·23-s − 0.200·25-s − 0.328·27-s − 1.28·29-s − 0.360·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.120 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.120 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6142888377\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6142888377\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + (-0.319 + 2.62i)T \) |
| good | 3 | \( 1 + 2.58T + 3T^{2} \) |
| 11 | \( 1 + 0.0961iT - 11T^{2} \) |
| 13 | \( 1 - 2.44iT - 13T^{2} \) |
| 17 | \( 1 - 2.17iT - 17T^{2} \) |
| 19 | \( 1 + 2.11T + 19T^{2} \) |
| 23 | \( 1 + 2.75iT - 23T^{2} \) |
| 29 | \( 1 + 6.89T + 29T^{2} \) |
| 31 | \( 1 + 2.00T + 31T^{2} \) |
| 37 | \( 1 - 3.95T + 37T^{2} \) |
| 41 | \( 1 + 6.44iT - 41T^{2} \) |
| 43 | \( 1 + 6.88iT - 43T^{2} \) |
| 47 | \( 1 - 12.4T + 47T^{2} \) |
| 53 | \( 1 - 13.6T + 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 61 | \( 1 - 8.32iT - 61T^{2} \) |
| 67 | \( 1 + 0.286iT - 67T^{2} \) |
| 71 | \( 1 - 12.7iT - 71T^{2} \) |
| 73 | \( 1 - 4.38iT - 73T^{2} \) |
| 79 | \( 1 - 9.94iT - 79T^{2} \) |
| 83 | \( 1 + 9.06T + 83T^{2} \) |
| 89 | \( 1 - 4.14iT - 89T^{2} \) |
| 97 | \( 1 - 16.3iT - 97T^{2} \) |
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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
−9.358851772214067354081253359365, −8.399498220030325406928403940633, −7.16099274349798500644874303252, −7.03256736156309087804963594784, −5.99907354589910813960581997032, −5.46178029596789051836557578554, −4.29648457940016911790346374093, −3.88606495857752011187933554050, −2.23648687447358413106497851463, −0.928765966831237855600611678153,
0.33580417947727745552800865030, 1.66468999862703774334859672078, 2.95477795985398191205647861107, 4.30834654789585089130000841654, 5.08615951446410362737374559698, 5.73381987731290307418077535327, 6.14807336263308135658728030506, 7.24806241685483685710974074574, 8.010094332582780122514804723868, 9.007318474985335579766366997193
