L(s) = 1 | + (−0.822 + 1.80i)3-s + (0.654 − 0.755i)5-s + (0.909 − 0.584i)7-s + (−1.91 − 2.20i)9-s + (0.822 + 1.80i)15-s + (0.304 + 2.11i)21-s + (0.989 − 0.142i)23-s + (−0.142 − 0.989i)25-s + (3.64 − 1.07i)27-s + (1.25 + 0.368i)29-s + (0.153 − 1.07i)35-s + (1.10 − 1.27i)41-s + (−0.234 + 0.512i)43-s − 2.91·45-s − 1.51·47-s + ⋯ |
L(s) = 1 | + (−0.822 + 1.80i)3-s + (0.654 − 0.755i)5-s + (0.909 − 0.584i)7-s + (−1.91 − 2.20i)9-s + (0.822 + 1.80i)15-s + (0.304 + 2.11i)21-s + (0.989 − 0.142i)23-s + (−0.142 − 0.989i)25-s + (3.64 − 1.07i)27-s + (1.25 + 0.368i)29-s + (0.153 − 1.07i)35-s + (1.10 − 1.27i)41-s + (−0.234 + 0.512i)43-s − 2.91·45-s − 1.51·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.779 - 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.779 - 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.053118455\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.053118455\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.654 + 0.755i)T \) |
| 23 | \( 1 + (-0.989 + 0.142i)T \) |
good | 3 | \( 1 + (0.822 - 1.80i)T + (-0.654 - 0.755i)T^{2} \) |
| 7 | \( 1 + (-0.909 + 0.584i)T + (0.415 - 0.909i)T^{2} \) |
| 11 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 13 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 17 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 19 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 29 | \( 1 + (-1.25 - 0.368i)T + (0.841 + 0.540i)T^{2} \) |
| 31 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 37 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 41 | \( 1 + (-1.10 + 1.27i)T + (-0.142 - 0.989i)T^{2} \) |
| 43 | \( 1 + (0.234 - 0.512i)T + (-0.654 - 0.755i)T^{2} \) |
| 47 | \( 1 + 1.51T + T^{2} \) |
| 53 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 59 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 61 | \( 1 + (-0.797 - 1.74i)T + (-0.654 + 0.755i)T^{2} \) |
| 67 | \( 1 + (-0.258 - 1.80i)T + (-0.959 + 0.281i)T^{2} \) |
| 71 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 73 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 79 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 83 | \( 1 + (1.19 + 1.37i)T + (-0.142 + 0.989i)T^{2} \) |
| 89 | \( 1 + (0.118 - 0.258i)T + (-0.654 - 0.755i)T^{2} \) |
| 97 | \( 1 + (0.142 + 0.989i)T^{2} \) |
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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
−9.722918378995223244245209065633, −8.835394045376140919134943003258, −8.410325868396466483082526319760, −6.98277088485354329763795798756, −5.95071665157127494233404927627, −5.28402761856935828625820787616, −4.63107370414096145123138461781, −4.14379592528252453638562532936, −2.89111473859549983144773386814, −1.07532664774796184467286191395,
1.28341904220946221199591964011, 2.15322099495091424901022504415, 2.93152000439744461002475282391, 4.95312728390143487465532094050, 5.44947851288468000673011513075, 6.50780388156842258647322958562, 6.64838445398247445233824243029, 7.80543764084296498491718221892, 8.160986520672252969594802446945, 9.236853943375714677388401141860