| L(s) = 1 | + (−1.02 + 0.970i)2-s + (−0.992 + 1.41i)3-s + (0.115 − 1.99i)4-s + (−1.30 + 1.81i)5-s + (−0.357 − 2.42i)6-s − 1.82·7-s + (1.81 + 2.16i)8-s + (−1.02 − 2.81i)9-s + (−0.412 − 3.13i)10-s + (0.247 + 0.761i)11-s + (2.71 + 2.14i)12-s + (−3.23 − 1.04i)13-s + (1.87 − 1.76i)14-s + (−1.27 − 3.65i)15-s + (−3.97 − 0.461i)16-s + (−0.461 − 0.335i)17-s + ⋯ |
| L(s) = 1 | + (−0.727 + 0.686i)2-s + (−0.573 + 0.819i)3-s + (0.0578 − 0.998i)4-s + (−0.585 + 0.810i)5-s + (−0.145 − 0.989i)6-s − 0.688·7-s + (0.643 + 0.765i)8-s + (−0.343 − 0.939i)9-s + (−0.130 − 0.991i)10-s + (0.0745 + 0.229i)11-s + (0.785 + 0.619i)12-s + (−0.896 − 0.291i)13-s + (0.500 − 0.472i)14-s + (−0.328 − 0.944i)15-s + (−0.993 − 0.115i)16-s + (−0.111 − 0.0813i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0283 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0283 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00273446 - 0.00265799i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00273446 - 0.00265799i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.02 - 0.970i)T \) |
| 3 | \( 1 + (0.992 - 1.41i)T \) |
| 5 | \( 1 + (1.30 - 1.81i)T \) |
| good | 7 | \( 1 + 1.82T + 7T^{2} \) |
| 11 | \( 1 + (-0.247 - 0.761i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (3.23 + 1.04i)T + (10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (0.461 + 0.335i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-4.02 + 5.54i)T + (-5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (1.85 - 0.602i)T + (18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (1.66 + 2.29i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (2.38 - 3.27i)T + (-9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-3.42 - 1.11i)T + (29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (10.0 + 3.26i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 5.23T + 43T^{2} \) |
| 47 | \( 1 + (6.91 + 9.51i)T + (-14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (6.16 - 4.47i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (2.59 - 7.98i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-4.03 - 12.4i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (7.80 + 5.66i)T + (20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (9.89 - 7.19i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (12.0 - 3.90i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-0.222 - 0.306i)T + (-24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (3.26 - 4.49i)T + (-25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-12.8 + 4.18i)T + (72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (0.331 + 0.455i)T + (-29.9 + 92.2i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.30706316605421476999234457016, −10.30078331237593784744701439312, −9.780602288879048805555291200703, −8.820081377254788108495024481997, −7.41417049572217652688818017320, −6.78225602344082048675234392414, −5.66234608730572653380202466037, −4.53620827622340561114244288332, −3.00191616424177663478755073738, −0.00376314017176353380298907110,
1.63305848273764871425024654589, 3.34666462074379223569457587067, 4.80734006957480096616206842506, 6.24561523410136164956274664861, 7.48940931134316684414235466696, 8.042284861634199606615604446023, 9.239246607355331875189661481353, 10.06496331338340483922468268586, 11.31011803202777018055504397993, 11.91534103315157284431879902669
