L(s) = 1 | + (0.965 + 0.258i)2-s + (0.866 + 0.499i)4-s + (0.866 − 0.5i)7-s + (0.707 + 0.707i)8-s + (−0.707 − 1.22i)11-s + (0.965 − 0.258i)14-s + (0.500 + 0.866i)16-s + (−0.366 − 1.36i)22-s + (0.707 − 1.22i)23-s + (0.5 + 0.866i)25-s + 28-s + (−1.22 + 0.707i)29-s + (0.258 + 0.965i)32-s + (−1.73 + i)43-s − 1.41i·44-s + ⋯ |
L(s) = 1 | + (0.965 + 0.258i)2-s + (0.866 + 0.499i)4-s + (0.866 − 0.5i)7-s + (0.707 + 0.707i)8-s + (−0.707 − 1.22i)11-s + (0.965 − 0.258i)14-s + (0.500 + 0.866i)16-s + (−0.366 − 1.36i)22-s + (0.707 − 1.22i)23-s + (0.5 + 0.866i)25-s + 28-s + (−1.22 + 0.707i)29-s + (0.258 + 0.965i)32-s + (−1.73 + i)43-s − 1.41i·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.322651991\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.322651991\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.965 - 0.258i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.866 + 0.5i)T \) |
good | 5 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 - 1.41iT - T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + 1.41T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T^{2} \) |
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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
−8.936958002920227838503864686074, −8.299719086465186400384290660763, −7.57353480373435655369143974282, −6.88627495058686476020821780490, −5.94055641940227577581744047088, −5.19191358919870325117677448862, −4.57564958106391441556241040839, −3.51619972645111178285888078069, −2.77139506310859130879761132176, −1.44989830467822619474232807458,
1.68050966117755348858029602773, 2.36793558688118244493277715232, 3.48148563076006588686472387375, 4.53423430464867560927688943461, 5.11498116857847309731498219371, 5.74874795846744145737328103995, 6.86571839809512271174419580578, 7.48490252598974582478391487901, 8.257137722591296678751911973624, 9.342101012732359123241153305906