L(s) = 1 | − 6.10i·2-s − 5.19·3-s − 21.2·4-s − 12.8·5-s + 31.7i·6-s + 4.61·7-s + 32.0i·8-s + 27·9-s + 78.4i·10-s + 159. i·11-s + 110.·12-s + 148. i·13-s − 28.1i·14-s + 66.7·15-s − 144.·16-s + 154.·17-s + ⋯ |
L(s) = 1 | − 1.52i·2-s − 0.577·3-s − 1.32·4-s − 0.513·5-s + 0.880i·6-s + 0.0942·7-s + 0.500i·8-s + 0.333·9-s + 0.784i·10-s + 1.31i·11-s + 0.766·12-s + 0.881i·13-s − 0.143i·14-s + 0.296·15-s − 0.564·16-s + 0.534·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.586 + 0.810i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.586 + 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(1.139575311\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.139575311\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 5.19T \) |
| 59 | \( 1 + (-2.04e3 - 2.82e3i)T \) |
good | 2 | \( 1 + 6.10iT - 16T^{2} \) |
| 5 | \( 1 + 12.8T + 625T^{2} \) |
| 7 | \( 1 - 4.61T + 2.40e3T^{2} \) |
| 11 | \( 1 - 159. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 148. iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 154.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 313.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 420. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 1.42e3T + 7.07e5T^{2} \) |
| 31 | \( 1 + 1.10e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 186. iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 435.T + 2.82e6T^{2} \) |
| 43 | \( 1 - 2.36e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 2.36e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 3.94e3T + 7.89e6T^{2} \) |
| 61 | \( 1 - 1.73e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 2.71e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 5.43e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 2.07e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 1.07e4T + 3.89e7T^{2} \) |
| 83 | \( 1 - 1.09e4iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 1.73e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 3.59e3iT - 8.85e7T^{2} \) |
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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
−11.95677672030056523457900133409, −11.07254225286977963556452797506, −10.03578992808756046134689044923, −9.405616563633726891800823215219, −7.81959114815346272351341120890, −6.59302447388165709798380864068, −4.80590090537965240032745041944, −3.98987761799690426641290555889, −2.40534371194816987493166760142, −1.02513787021111082251684641593,
0.59756065813692034207403646600, 3.47012638101266496076157841417, 5.11820206451958899679763640337, 5.78014233754393167843932096527, 6.90857283271205872230412673606, 7.909374438236446576822948963940, 8.609212448971439485106048231402, 10.07155941139016971009710965898, 11.26443560765922028742004605654, 12.12809438498601468719672712381