| L(s) = 1 | + 3-s − 3.21i·5-s + 9-s − 1.36i·11-s + 2.93i·13-s − 3.21i·15-s − 6.91i·17-s − 7.35·19-s − 3.62i·23-s − 5.33·25-s + 27-s − 1.11·29-s − 8.70·31-s − 1.36i·33-s + 7.63·37-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.43i·5-s + 0.333·9-s − 0.412i·11-s + 0.812i·13-s − 0.830i·15-s − 1.67i·17-s − 1.68·19-s − 0.756i·23-s − 1.06·25-s + 0.192·27-s − 0.206·29-s − 1.56·31-s − 0.237i·33-s + 1.25·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.810 + 0.585i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.810 + 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.450291544\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.450291544\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 3.21iT - 5T^{2} \) |
| 11 | \( 1 + 1.36iT - 11T^{2} \) |
| 13 | \( 1 - 2.93iT - 13T^{2} \) |
| 17 | \( 1 + 6.91iT - 17T^{2} \) |
| 19 | \( 1 + 7.35T + 19T^{2} \) |
| 23 | \( 1 + 3.62iT - 23T^{2} \) |
| 29 | \( 1 + 1.11T + 29T^{2} \) |
| 31 | \( 1 + 8.70T + 31T^{2} \) |
| 37 | \( 1 - 7.63T + 37T^{2} \) |
| 41 | \( 1 - 0.833iT - 41T^{2} \) |
| 43 | \( 1 - 4.82iT - 43T^{2} \) |
| 47 | \( 1 + 2.95T + 47T^{2} \) |
| 53 | \( 1 + 4.57T + 53T^{2} \) |
| 59 | \( 1 - 14.0T + 59T^{2} \) |
| 61 | \( 1 + 11.0iT - 61T^{2} \) |
| 67 | \( 1 + 12.1iT - 67T^{2} \) |
| 71 | \( 1 - 12.6iT - 71T^{2} \) |
| 73 | \( 1 - 6.49iT - 73T^{2} \) |
| 79 | \( 1 - 7.79iT - 79T^{2} \) |
| 83 | \( 1 + 8.87T + 83T^{2} \) |
| 89 | \( 1 + 12.6iT - 89T^{2} \) |
| 97 | \( 1 + 13.8iT - 97T^{2} \) |
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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
−8.644009054889098590901249375630, −8.211945544190292687281050354908, −7.20976396225331089630848027692, −6.39735406662885571537251378108, −5.33937802212861164430021759447, −4.58549257017784938954961304913, −3.99216346497277557390629958315, −2.68800276432933632743689611105, −1.71158374473677723181529034224, −0.42514051841251673791421299740,
1.80797982860474194924560574346, 2.58469725817770469206713267874, 3.59538953167799741886727697609, 4.12430351884061123253404688471, 5.55589613688159103634580954526, 6.30152979187178931612407832753, 7.02663804705038169715366879907, 7.75835200694959223257470830944, 8.401970401033194951549709224248, 9.289669192521966781455399673496
