L(s) = 1 | + (0.294 − 0.955i)3-s + (0.988 − 0.149i)4-s + (0.563 + 0.826i)7-s + (−0.826 − 0.563i)9-s + (0.149 − 0.988i)12-s + (0.636 − 1.32i)13-s + (0.955 − 0.294i)16-s + (0.826 + 1.43i)19-s + (0.955 − 0.294i)21-s + (−0.781 + 0.623i)27-s + (0.680 + 0.733i)28-s + (−0.623 + 1.07i)31-s + (−0.900 − 0.433i)36-s + (0.108 − 0.722i)37-s + (−1.07 − 0.997i)39-s + ⋯ |
L(s) = 1 | + (0.294 − 0.955i)3-s + (0.988 − 0.149i)4-s + (0.563 + 0.826i)7-s + (−0.826 − 0.563i)9-s + (0.149 − 0.988i)12-s + (0.636 − 1.32i)13-s + (0.955 − 0.294i)16-s + (0.826 + 1.43i)19-s + (0.955 − 0.294i)21-s + (−0.781 + 0.623i)27-s + (0.680 + 0.733i)28-s + (−0.623 + 1.07i)31-s + (−0.900 − 0.433i)36-s + (0.108 − 0.722i)37-s + (−1.07 − 0.997i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.626 + 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.626 + 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.965576997\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.965576997\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.294 + 0.955i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.563 - 0.826i)T \) |
good | 2 | \( 1 + (-0.988 + 0.149i)T^{2} \) |
| 11 | \( 1 + (-0.365 + 0.930i)T^{2} \) |
| 13 | \( 1 + (-0.636 + 1.32i)T + (-0.623 - 0.781i)T^{2} \) |
| 17 | \( 1 + (-0.733 + 0.680i)T^{2} \) |
| 19 | \( 1 + (-0.826 - 1.43i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.733 - 0.680i)T^{2} \) |
| 29 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 31 | \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.108 + 0.722i)T + (-0.955 - 0.294i)T^{2} \) |
| 41 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 43 | \( 1 + (1.75 + 0.400i)T + (0.900 + 0.433i)T^{2} \) |
| 47 | \( 1 + (-0.988 + 0.149i)T^{2} \) |
| 53 | \( 1 + (0.955 - 0.294i)T^{2} \) |
| 59 | \( 1 + (-0.0747 - 0.997i)T^{2} \) |
| 61 | \( 1 + (-1.95 - 0.294i)T + (0.955 + 0.294i)T^{2} \) |
| 67 | \( 1 + (1.65 + 0.955i)T + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 73 | \( 1 + (0.728 + 0.0546i)T + (0.988 + 0.149i)T^{2} \) |
| 79 | \( 1 + (0.988 + 1.71i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.623 - 0.781i)T^{2} \) |
| 89 | \( 1 + (-0.365 - 0.930i)T^{2} \) |
| 97 | \( 1 - 0.149iT - T^{2} \) |
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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
−8.345032314914893306557444067793, −7.86395497887011076186724958817, −7.25178001945109497352819627976, −6.35141852375956223036730154975, −5.69175533314946324777841075505, −5.28457066110045806662681374446, −3.51923831889450914333401067450, −2.99114441166055704819776649375, −1.94868084683447675919625845531, −1.29159883436679084329789261202,
1.45751313645617828397654049947, 2.48942040418828431417264304109, 3.41451888279491963328730514603, 4.16147193690037436510421490913, 4.88284484108165708809668941561, 5.79519287739370142790497781696, 6.78550730553175025640354706409, 7.24648905633882935418608446707, 8.159177448601300809780023508958, 8.755985028603557278271076490103