L(s) = 1 | + (−0.869 + 0.329i)5-s + (0.354 + 0.935i)9-s + (0.213 + 0.112i)13-s + (1.45 + 1.28i)17-s + (−0.101 + 0.0902i)25-s + (0.0854 + 0.704i)29-s + (0.850 − 1.23i)37-s + (−1.31 − 0.159i)41-s + (−0.616 − 0.695i)45-s + (0.885 − 0.464i)49-s + (−0.568 + 0.822i)53-s + (−1.24 − 1.39i)61-s + (−0.222 − 0.0270i)65-s + (1.09 − 1.23i)73-s + (−0.748 + 0.663i)81-s + ⋯ |
L(s) = 1 | + (−0.869 + 0.329i)5-s + (0.354 + 0.935i)9-s + (0.213 + 0.112i)13-s + (1.45 + 1.28i)17-s + (−0.101 + 0.0902i)25-s + (0.0854 + 0.704i)29-s + (0.850 − 1.23i)37-s + (−1.31 − 0.159i)41-s + (−0.616 − 0.695i)45-s + (0.885 − 0.464i)49-s + (−0.568 + 0.822i)53-s + (−1.24 − 1.39i)61-s + (−0.222 − 0.0270i)65-s + (1.09 − 1.23i)73-s + (−0.748 + 0.663i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.638 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.638 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8752980454\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8752980454\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 53 | \( 1 + (0.568 - 0.822i)T \) |
good | 3 | \( 1 + (-0.354 - 0.935i)T^{2} \) |
| 5 | \( 1 + (0.869 - 0.329i)T + (0.748 - 0.663i)T^{2} \) |
| 7 | \( 1 + (-0.885 + 0.464i)T^{2} \) |
| 11 | \( 1 + (0.970 + 0.239i)T^{2} \) |
| 13 | \( 1 + (-0.213 - 0.112i)T + (0.568 + 0.822i)T^{2} \) |
| 17 | \( 1 + (-1.45 - 1.28i)T + (0.120 + 0.992i)T^{2} \) |
| 19 | \( 1 + (0.568 + 0.822i)T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + (-0.0854 - 0.704i)T + (-0.970 + 0.239i)T^{2} \) |
| 31 | \( 1 + (-0.970 + 0.239i)T^{2} \) |
| 37 | \( 1 + (-0.850 + 1.23i)T + (-0.354 - 0.935i)T^{2} \) |
| 41 | \( 1 + (1.31 + 0.159i)T + (0.970 + 0.239i)T^{2} \) |
| 43 | \( 1 + (0.354 - 0.935i)T^{2} \) |
| 47 | \( 1 + (0.748 + 0.663i)T^{2} \) |
| 59 | \( 1 + (0.748 + 0.663i)T^{2} \) |
| 61 | \( 1 + (1.24 + 1.39i)T + (-0.120 + 0.992i)T^{2} \) |
| 67 | \( 1 + (0.568 - 0.822i)T^{2} \) |
| 71 | \( 1 + (-0.354 + 0.935i)T^{2} \) |
| 73 | \( 1 + (-1.09 + 1.23i)T + (-0.120 - 0.992i)T^{2} \) |
| 79 | \( 1 + (0.885 + 0.464i)T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (1.12 + 0.992i)T + (0.120 + 0.992i)T^{2} \) |
| 97 | \( 1 + (0.251 + 0.663i)T + (-0.748 + 0.663i)T^{2} \) |
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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
−10.62235807130440967555513668653, −9.826632784596983975728727974038, −8.654414252104686702116543000179, −7.82599297506609514646595434297, −7.37131254484353491092991594340, −6.17008099634301661852611298056, −5.18914794154091013173205548889, −4.07608322116015739548470649927, −3.25998016055265053412069370742, −1.72124871556899464390110899817,
1.00530400678433386647928721262, 2.96778221645688441423434739457, 3.87841713882613420151407430742, 4.82896415695533935372620066430, 5.92127570108007526984514821646, 6.95359483551933804302489372891, 7.76453513677304088492935608752, 8.494201901890698869390966569348, 9.561008610711056942800282899590, 10.05445173840467564620341414323