L(s) = 1 | + (−0.187 + 0.982i)3-s + (0.309 + 0.951i)4-s + (1.69 − 0.435i)7-s + (−0.929 − 0.368i)9-s + (−0.992 + 0.125i)12-s + (−1.62 − 0.895i)13-s + (−0.809 + 0.587i)16-s + (0.0388 + 0.119i)19-s + (0.110 + 1.74i)21-s + (0.728 + 0.684i)25-s + (0.535 − 0.844i)27-s + (0.939 + 1.47i)28-s + (−1.23 − 1.49i)31-s + (0.0627 − 0.998i)36-s + (−0.328 − 0.180i)37-s + ⋯ |
L(s) = 1 | + (−0.187 + 0.982i)3-s + (0.309 + 0.951i)4-s + (1.69 − 0.435i)7-s + (−0.929 − 0.368i)9-s + (−0.992 + 0.125i)12-s + (−1.62 − 0.895i)13-s + (−0.809 + 0.587i)16-s + (0.0388 + 0.119i)19-s + (0.110 + 1.74i)21-s + (0.728 + 0.684i)25-s + (0.535 − 0.844i)27-s + (0.939 + 1.47i)28-s + (−1.23 − 1.49i)31-s + (0.0627 − 0.998i)36-s + (−0.328 − 0.180i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 453 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.311 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 453 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.311 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9194745362\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9194745362\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.187 - 0.982i)T \) |
| 151 | \( 1 + (0.809 - 0.587i)T \) |
good | 2 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 5 | \( 1 + (-0.728 - 0.684i)T^{2} \) |
| 7 | \( 1 + (-1.69 + 0.435i)T + (0.876 - 0.481i)T^{2} \) |
| 11 | \( 1 + (0.929 - 0.368i)T^{2} \) |
| 13 | \( 1 + (1.62 + 0.895i)T + (0.535 + 0.844i)T^{2} \) |
| 17 | \( 1 + (-0.876 - 0.481i)T^{2} \) |
| 19 | \( 1 + (-0.0388 - 0.119i)T + (-0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 29 | \( 1 + (0.637 + 0.770i)T^{2} \) |
| 31 | \( 1 + (1.23 + 1.49i)T + (-0.187 + 0.982i)T^{2} \) |
| 37 | \( 1 + (0.328 + 0.180i)T + (0.535 + 0.844i)T^{2} \) |
| 41 | \( 1 + (-0.0627 + 0.998i)T^{2} \) |
| 43 | \( 1 + (0.824 + 0.211i)T + (0.876 + 0.481i)T^{2} \) |
| 47 | \( 1 + (-0.0627 + 0.998i)T^{2} \) |
| 53 | \( 1 + (-0.968 - 0.248i)T^{2} \) |
| 59 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (-0.303 - 1.58i)T + (-0.929 + 0.368i)T^{2} \) |
| 67 | \( 1 + (-1.84 + 0.730i)T + (0.728 - 0.684i)T^{2} \) |
| 71 | \( 1 + (-0.876 + 0.481i)T^{2} \) |
| 73 | \( 1 + (-0.598 + 0.153i)T + (0.876 - 0.481i)T^{2} \) |
| 79 | \( 1 + (0.929 - 1.12i)T + (-0.187 - 0.982i)T^{2} \) |
| 83 | \( 1 + (0.992 - 0.125i)T^{2} \) |
| 89 | \( 1 + (0.992 - 0.125i)T^{2} \) |
| 97 | \( 1 + (-1.50 + 0.595i)T + (0.728 - 0.684i)T^{2} \) |
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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
−11.35554635179758645903182897446, −10.77274488896630079813755065424, −9.782688620188235773230093608296, −8.668926262386278208592422676884, −7.85160877369549239595562156120, −7.18551477364581972225505368721, −5.43966536860211811159081041233, −4.72884897902447948525066446093, −3.71597531136328624709975116924, −2.38207654027055214659021692046,
1.59654921021572787441263118420, 2.38464112699019949780195248097, 4.89816415683581011871239378575, 5.23418503103595976536744316554, 6.60917444140040560921785982401, 7.29216041204330865167419584596, 8.310760568806795300790670984177, 9.226469027443271036353700275322, 10.46142256910348050449046310618, 11.29142396929123064323416227233