L(s) = 1 | − 103. i·5-s − 1.65e3·13-s − 6.21e3i·17-s + 4.96e3·25-s + 3.56e3i·29-s − 5.55e4·37-s + 1.67e4i·41-s − 1.17e5·49-s − 2.30e5i·53-s − 2.34e5·61-s + 1.70e5i·65-s + 6.50e5·73-s − 6.41e5·85-s + 7.67e5i·89-s + 1.07e6·97-s + ⋯ |
L(s) = 1 | − 0.825i·5-s − 0.753·13-s − 1.26i·17-s + 0.317·25-s + 0.146i·29-s − 1.09·37-s + 0.242i·41-s − 49-s − 1.54i·53-s − 1.03·61-s + 0.622i·65-s + 1.67·73-s − 1.04·85-s + 1.08i·89-s + 1.18·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(0.09329593060\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.09329593060\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 103. iT - 1.56e4T^{2} \) |
| 7 | \( 1 + 1.17e5T^{2} \) |
| 11 | \( 1 - 1.77e6T^{2} \) |
| 13 | \( 1 + 1.65e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + 6.21e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 4.70e7T^{2} \) |
| 23 | \( 1 - 1.48e8T^{2} \) |
| 29 | \( 1 - 3.56e3iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 8.87e8T^{2} \) |
| 37 | \( 1 + 5.55e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 1.67e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 6.32e9T^{2} \) |
| 47 | \( 1 - 1.07e10T^{2} \) |
| 53 | \( 1 + 2.30e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 4.21e10T^{2} \) |
| 61 | \( 1 + 2.34e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + 9.04e10T^{2} \) |
| 71 | \( 1 - 1.28e11T^{2} \) |
| 73 | \( 1 - 6.50e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 2.43e11T^{2} \) |
| 83 | \( 1 - 3.26e11T^{2} \) |
| 89 | \( 1 - 7.67e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 1.07e6T + 8.32e11T^{2} \) |
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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
−9.883850238927567788295968170457, −9.236485369573695849615189280423, −8.377614774582335277076538103104, −7.43412565774805401372519800697, −6.54663847074108616080283997854, −5.19733825499674766625518509855, −4.78207319795807977791697792850, −3.42864118816827463727848090644, −2.25170215900847200047149413271, −1.00142957499697520391333921063,
0.02009874049664592055004052627, 1.58074804641242040579825958747, 2.68679126481278772277016208048, 3.66251410533916320432010938252, 4.79133552674380420816304260919, 5.93461729155541545814879542640, 6.77403057071346917611064601380, 7.60521057102904600263326165761, 8.541707784848133427601742590589, 9.561960090031519415400156754267