L(s) = 1 | + (0.692 + 0.721i)3-s + (0.568 − 0.822i)4-s + (0.987 − 0.160i)7-s + (−0.0402 + 0.999i)9-s + (0.987 − 0.160i)12-s + (−0.5 + 0.866i)13-s + (−0.354 − 0.935i)16-s + (0.845 − 1.46i)19-s + (0.799 + 0.600i)21-s + (0.948 − 0.316i)25-s + (−0.748 + 0.663i)27-s + (0.428 − 0.903i)28-s + (−0.227 − 1.11i)31-s + (0.799 + 0.600i)36-s + (−1.19 + 1.06i)37-s + ⋯ |
L(s) = 1 | + (0.692 + 0.721i)3-s + (0.568 − 0.822i)4-s + (0.987 − 0.160i)7-s + (−0.0402 + 0.999i)9-s + (0.987 − 0.160i)12-s + (−0.5 + 0.866i)13-s + (−0.354 − 0.935i)16-s + (0.845 − 1.46i)19-s + (0.799 + 0.600i)21-s + (0.948 − 0.316i)25-s + (−0.748 + 0.663i)27-s + (0.428 − 0.903i)28-s + (−0.227 − 1.11i)31-s + (0.799 + 0.600i)36-s + (−1.19 + 1.06i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.061578310\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.061578310\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.692 - 0.721i)T \) |
| 7 | \( 1 + (-0.987 + 0.160i)T \) |
| 13 | \( 1 + (0.5 - 0.866i)T \) |
good | 2 | \( 1 + (-0.568 + 0.822i)T^{2} \) |
| 5 | \( 1 + (-0.948 + 0.316i)T^{2} \) |
| 11 | \( 1 + (0.996 - 0.0804i)T^{2} \) |
| 17 | \( 1 + (0.970 + 0.239i)T^{2} \) |
| 19 | \( 1 + (-0.845 + 1.46i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (0.996 + 0.0804i)T^{2} \) |
| 31 | \( 1 + (0.227 + 1.11i)T + (-0.919 + 0.391i)T^{2} \) |
| 37 | \( 1 + (1.19 - 1.06i)T + (0.120 - 0.992i)T^{2} \) |
| 41 | \( 1 + (0.845 + 0.534i)T^{2} \) |
| 43 | \( 1 + (0.319 - 1.56i)T + (-0.919 - 0.391i)T^{2} \) |
| 47 | \( 1 + (0.632 + 0.774i)T^{2} \) |
| 53 | \( 1 + (-0.278 - 0.960i)T^{2} \) |
| 59 | \( 1 + (0.748 + 0.663i)T^{2} \) |
| 61 | \( 1 + (0.788 + 0.336i)T + (0.692 + 0.721i)T^{2} \) |
| 67 | \( 1 + (0.641 - 1.35i)T + (-0.632 - 0.774i)T^{2} \) |
| 71 | \( 1 + (0.0402 - 0.999i)T^{2} \) |
| 73 | \( 1 + (-1.06 + 0.676i)T + (0.428 - 0.903i)T^{2} \) |
| 79 | \( 1 + (0.832 - 1.75i)T + (-0.632 - 0.774i)T^{2} \) |
| 83 | \( 1 + (-0.885 - 0.464i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.632 + 0.774i)T + (-0.200 - 0.979i)T^{2} \) |
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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
−8.903161148313126206230626195149, −8.059507631554052573769685100897, −7.26461786074654714236388641795, −6.69896004160341741042359499158, −5.52140312921359039771905809270, −4.82240320533303204399084265671, −4.41961893795964880257204714987, −3.06014852805979117934130902875, −2.32147903190398084596676680562, −1.36259371259280979940299842056,
1.42900657095257070058935618021, 2.20655273541170752976243026315, 3.19182324840716385166527955080, 3.71973011097182950979113578799, 5.00715573385911623102963100801, 5.78263593604897905901518785208, 6.80889639585564880178710747316, 7.51785000934028170840117672918, 7.79497888092837442334682909852, 8.610852167076044717621001174736