L(s) = 1 | + (−1.39 + 0.212i)2-s + (1.67 − 0.428i)3-s + (1.90 − 0.594i)4-s + 1.58·5-s + (−2.25 + 0.955i)6-s + 2.65i·7-s + (−2.54 + 1.23i)8-s + (2.63 − 1.43i)9-s + (−2.20 + 0.336i)10-s + 1.71i·11-s + (2.95 − 1.81i)12-s + 4.09i·13-s + (−0.565 − 3.71i)14-s + (2.65 − 0.676i)15-s + (3.29 − 2.27i)16-s + 5.87i·17-s + ⋯ |
L(s) = 1 | + (−0.988 + 0.150i)2-s + (0.968 − 0.247i)3-s + (0.954 − 0.297i)4-s + 0.706·5-s + (−0.920 + 0.390i)6-s + 1.00i·7-s + (−0.899 + 0.437i)8-s + (0.877 − 0.479i)9-s + (−0.698 + 0.106i)10-s + 0.517i·11-s + (0.851 − 0.524i)12-s + 1.13i·13-s + (−0.151 − 0.993i)14-s + (0.684 − 0.174i)15-s + (0.823 − 0.567i)16-s + 1.42i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.646i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.763 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.38259 + 0.506765i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.38259 + 0.506765i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.39 - 0.212i)T \) |
| 3 | \( 1 + (-1.67 + 0.428i)T \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 - 1.58T + 5T^{2} \) |
| 7 | \( 1 - 2.65iT - 7T^{2} \) |
| 11 | \( 1 - 1.71iT - 11T^{2} \) |
| 13 | \( 1 - 4.09iT - 13T^{2} \) |
| 17 | \( 1 - 5.87iT - 17T^{2} \) |
| 19 | \( 1 + 1.13T + 19T^{2} \) |
| 29 | \( 1 - 3.33T + 29T^{2} \) |
| 31 | \( 1 + 9.90iT - 31T^{2} \) |
| 37 | \( 1 + 4.56iT - 37T^{2} \) |
| 41 | \( 1 + 9.73iT - 41T^{2} \) |
| 43 | \( 1 + 6.27T + 43T^{2} \) |
| 47 | \( 1 - 2.62T + 47T^{2} \) |
| 53 | \( 1 - 8.73T + 53T^{2} \) |
| 59 | \( 1 - 1.29iT - 59T^{2} \) |
| 61 | \( 1 - 1.42iT - 61T^{2} \) |
| 67 | \( 1 + 8.90T + 67T^{2} \) |
| 71 | \( 1 - 5.40T + 71T^{2} \) |
| 73 | \( 1 - 8.08T + 73T^{2} \) |
| 79 | \( 1 - 4.26iT - 79T^{2} \) |
| 83 | \( 1 + 16.3iT - 83T^{2} \) |
| 89 | \( 1 - 7.01iT - 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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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
−10.49280038270618915093005903874, −9.728017868896891184804062805382, −9.051966517513182170544064486611, −8.471730949455593665108541590037, −7.49708898331544509009297846130, −6.49170163352601166487924541172, −5.74715153317892517055403819777, −3.98974321390806129568923725756, −2.30049268658029241611039687737, −1.88410454157461886360082622909,
1.12840210692917411610226807313, 2.63847461847538701709254521549, 3.48399193013034333472630955563, 5.03616481480745937465016681406, 6.48366382802965259358741910115, 7.38062957719481912107182925117, 8.162726770236947890061313885704, 8.944673894186256100591776127180, 9.997220889879492960481709737221, 10.20086267821146540004522078145