L(s) = 1 | + (0.876 + 0.481i)2-s + (0.535 + 0.844i)4-s + (−1.62 + 0.895i)5-s + (0.0627 + 0.998i)8-s + (0.968 + 0.248i)9-s − 1.85·10-s + (0.371 − 1.94i)13-s + (−0.425 + 0.904i)16-s + (−0.5 + 0.363i)17-s + (0.728 + 0.684i)18-s + (−1.62 − 0.895i)20-s + (1.31 − 2.07i)25-s + (1.26 − 1.52i)26-s + (0.303 − 1.58i)29-s + (−0.809 + 0.587i)32-s + ⋯ |
L(s) = 1 | + (0.876 + 0.481i)2-s + (0.535 + 0.844i)4-s + (−1.62 + 0.895i)5-s + (0.0627 + 0.998i)8-s + (0.968 + 0.248i)9-s − 1.85·10-s + (0.371 − 1.94i)13-s + (−0.425 + 0.904i)16-s + (−0.5 + 0.363i)17-s + (0.728 + 0.684i)18-s + (−1.62 − 0.895i)20-s + (1.31 − 2.07i)25-s + (1.26 − 1.52i)26-s + (0.303 − 1.58i)29-s + (−0.809 + 0.587i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 404 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 404 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.115141029\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.115141029\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.876 - 0.481i)T \) |
| 101 | \( 1 + (-0.535 - 0.844i)T \) |
good | 3 | \( 1 + (-0.968 - 0.248i)T^{2} \) |
| 5 | \( 1 + (1.62 - 0.895i)T + (0.535 - 0.844i)T^{2} \) |
| 7 | \( 1 + (0.929 - 0.368i)T^{2} \) |
| 11 | \( 1 + (-0.876 - 0.481i)T^{2} \) |
| 13 | \( 1 + (-0.371 + 1.94i)T + (-0.929 - 0.368i)T^{2} \) |
| 17 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (0.637 - 0.770i)T^{2} \) |
| 23 | \( 1 + (-0.0627 + 0.998i)T^{2} \) |
| 29 | \( 1 + (-0.303 + 1.58i)T + (-0.929 - 0.368i)T^{2} \) |
| 31 | \( 1 + (0.929 - 0.368i)T^{2} \) |
| 37 | \( 1 + (1.06 - 0.134i)T + (0.968 - 0.248i)T^{2} \) |
| 41 | \( 1 + (-0.688 - 0.500i)T + (0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 + (0.187 + 0.982i)T^{2} \) |
| 47 | \( 1 + (0.187 - 0.982i)T^{2} \) |
| 53 | \( 1 + (0.200 - 0.316i)T + (-0.425 - 0.904i)T^{2} \) |
| 59 | \( 1 + (0.637 + 0.770i)T^{2} \) |
| 61 | \( 1 + (-0.0672 - 0.106i)T + (-0.425 + 0.904i)T^{2} \) |
| 67 | \( 1 + (-0.968 + 0.248i)T^{2} \) |
| 71 | \( 1 + (-0.968 - 0.248i)T^{2} \) |
| 73 | \( 1 + (0.929 - 0.872i)T + (0.0627 - 0.998i)T^{2} \) |
| 79 | \( 1 + (-0.0627 - 0.998i)T^{2} \) |
| 83 | \( 1 + (-0.0627 - 0.998i)T^{2} \) |
| 89 | \( 1 + (0.824 + 1.75i)T + (-0.637 + 0.770i)T^{2} \) |
| 97 | \( 1 + (0.200 + 0.316i)T + (-0.425 + 0.904i)T^{2} \) |
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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
−11.74881890703525582755679497915, −10.93197892990552104677341674081, −10.26269268085724523640065374062, −8.307330681233399327469666476992, −7.80956347671629043466096606312, −7.06407759996346664812974838838, −6.04105645860618605559063321616, −4.63507153234087271409437330399, −3.78558349251806572454379672040, −2.85356446163337558246027193514,
1.53361378080098745944304798823, 3.57718833767186273170724443687, 4.30530615264840712755928236577, 4.96101602241353142066452480169, 6.70220235889685180642422297904, 7.27441964695970020092170709497, 8.698599450090189328988532489877, 9.428310866548496251284053152644, 10.80928903092427521320567539526, 11.52970463501815706213698176682