L(s) = 1 | + (1 − 1.73i)3-s + 4·7-s + (−0.499 − 0.866i)9-s − 3·11-s + (3 + 5.19i)13-s + (1 − 1.73i)17-s + (3.5 + 2.59i)19-s + (4 − 6.92i)21-s + (2 + 3.46i)23-s + 4.00·27-s + (−0.5 − 0.866i)29-s − 5·31-s + (−3 + 5.19i)33-s + 4·37-s + 12·39-s + ⋯ |
L(s) = 1 | + (0.577 − 0.999i)3-s + 1.51·7-s + (−0.166 − 0.288i)9-s − 0.904·11-s + (0.832 + 1.44i)13-s + (0.242 − 0.420i)17-s + (0.802 + 0.596i)19-s + (0.872 − 1.51i)21-s + (0.417 + 0.722i)23-s + 0.769·27-s + (−0.0928 − 0.160i)29-s − 0.898·31-s + (−0.522 + 0.904i)33-s + 0.657·37-s + 1.92·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.661556173\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.661556173\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-3.5 - 2.59i)T \) |
good | 3 | \( 1 + (-1 + 1.73i)T + (-1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 - 4T + 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + (-3 - 5.19i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 5T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + (1 - 1.73i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.5 + 0.866i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (7 + 12.1i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (7.5 - 12.9i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-6 + 10.3i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 16T + 83T^{2} \) |
| 89 | \( 1 + (8.5 + 14.7i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-6 + 10.3i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.885271539108395495634580380864, −8.262525055877581624746629679982, −7.54812121296920486526209381431, −7.20077439042336357277017919332, −5.98977008679118641934874384442, −5.11702550620692209216151035385, −4.26836777191941391460572853982, −3.02160327713235391472926172150, −1.88163309739496665316701301808, −1.35554462950371377971177577490,
1.08844407421819833641327426099, 2.55875957768212249296486296401, 3.40930429422636265560328362352, 4.35293500856012483902849609656, 5.16408250055960883457152531609, 5.69030909251564658771053480452, 7.14660415200485460840162573864, 8.028011200385701199778456805967, 8.403553996277668332390904360092, 9.185966347256608838486731586542