L(s) = 1 | − 2.49·2-s + i·3-s + 4.23·4-s + (2.05 − 0.879i)5-s − 2.49i·6-s + 3.27i·7-s − 5.57·8-s − 9-s + (−5.13 + 2.19i)10-s − 2.24·11-s + 4.23i·12-s − 3.94·13-s − 8.18i·14-s + (0.879 + 2.05i)15-s + 5.45·16-s − 5.46·17-s + ⋯ |
L(s) = 1 | − 1.76·2-s + 0.577i·3-s + 2.11·4-s + (0.919 − 0.393i)5-s − 1.01i·6-s + 1.23i·7-s − 1.97·8-s − 0.333·9-s + (−1.62 + 0.694i)10-s − 0.678·11-s + 1.22i·12-s − 1.09·13-s − 2.18i·14-s + (0.227 + 0.530i)15-s + 1.36·16-s − 1.32·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.947 - 0.320i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.947 - 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0545859 + 0.331496i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0545859 + 0.331496i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 5 | \( 1 + (-2.05 + 0.879i)T \) |
| 37 | \( 1 + (-4.05 + 4.52i)T \) |
good | 2 | \( 1 + 2.49T + 2T^{2} \) |
| 7 | \( 1 - 3.27iT - 7T^{2} \) |
| 11 | \( 1 + 2.24T + 11T^{2} \) |
| 13 | \( 1 + 3.94T + 13T^{2} \) |
| 17 | \( 1 + 5.46T + 17T^{2} \) |
| 19 | \( 1 - 4.91iT - 19T^{2} \) |
| 23 | \( 1 - 0.293T + 23T^{2} \) |
| 29 | \( 1 + 4.42iT - 29T^{2} \) |
| 31 | \( 1 - 9.50iT - 31T^{2} \) |
| 41 | \( 1 + 7.53T + 41T^{2} \) |
| 43 | \( 1 + 9.74T + 43T^{2} \) |
| 47 | \( 1 + 5.70iT - 47T^{2} \) |
| 53 | \( 1 - 14.1iT - 53T^{2} \) |
| 59 | \( 1 - 4.15iT - 59T^{2} \) |
| 61 | \( 1 + 1.76iT - 61T^{2} \) |
| 67 | \( 1 + 2.23iT - 67T^{2} \) |
| 71 | \( 1 + 1.23T + 71T^{2} \) |
| 73 | \( 1 - 2.78iT - 73T^{2} \) |
| 79 | \( 1 + 7.60iT - 79T^{2} \) |
| 83 | \( 1 - 7.35iT - 83T^{2} \) |
| 89 | \( 1 - 11.7iT - 89T^{2} \) |
| 97 | \( 1 - 8.37T + 97T^{2} \) |
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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
−10.63712949474171573939299840565, −10.10646134017514338862383132281, −9.347850930579635405815790236380, −8.772324993331208083368272228558, −8.082596571548182787833282471091, −6.81944594970689744664802955581, −5.82746778108835458106200178032, −4.89994204908134545243423720772, −2.69251455545429692079121817475, −1.92323086962168798666967308519,
0.30989185724812090434399690990, 1.85699231014239415419775265719, 2.77966959396842436630355455742, 4.90144246459983405901028862129, 6.49840256987325367827470641882, 6.97593497706787032536466956389, 7.68193993976573845596226521241, 8.659119359744687859089855085088, 9.644461629420999900058435631666, 10.15341925159272204291502069619