| L(s) = 1 | − 1.53·5-s + (0.414 − 2.61i)7-s + 0.585i·11-s + 2.16i·13-s − 5.86·17-s − 5.22i·19-s − 2.24i·23-s − 2.65·25-s + 5.41i·29-s − 4.32i·31-s + (−0.634 + 4i)35-s + 4·37-s − 8.92·41-s − 10.4·43-s − 7.39·47-s + ⋯ |
| L(s) = 1 | − 0.684·5-s + (0.156 − 0.987i)7-s + 0.176i·11-s + 0.600i·13-s − 1.42·17-s − 1.19i·19-s − 0.467i·23-s − 0.531·25-s + 1.00i·29-s − 0.777i·31-s + (−0.107 + 0.676i)35-s + 0.657·37-s − 1.39·41-s − 1.59·43-s − 1.07·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 + 0.442i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.896 + 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4810617634\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4810617634\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.414 + 2.61i)T \) |
| good | 5 | \( 1 + 1.53T + 5T^{2} \) |
| 11 | \( 1 - 0.585iT - 11T^{2} \) |
| 13 | \( 1 - 2.16iT - 13T^{2} \) |
| 17 | \( 1 + 5.86T + 17T^{2} \) |
| 19 | \( 1 + 5.22iT - 19T^{2} \) |
| 23 | \( 1 + 2.24iT - 23T^{2} \) |
| 29 | \( 1 - 5.41iT - 29T^{2} \) |
| 31 | \( 1 + 4.32iT - 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + 8.92T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 + 7.39T + 47T^{2} \) |
| 53 | \( 1 + 5.41iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 9.65T + 67T^{2} \) |
| 71 | \( 1 + 4.58iT - 71T^{2} \) |
| 73 | \( 1 + 12.6iT - 73T^{2} \) |
| 79 | \( 1 - 2.34T + 79T^{2} \) |
| 83 | \( 1 + 13.5T + 83T^{2} \) |
| 89 | \( 1 + 5.86T + 89T^{2} \) |
| 97 | \( 1 + 8.28iT - 97T^{2} \) |
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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
−9.617624096074408835714448071269, −8.713401095882491135532029942028, −7.946873465704492659578605930478, −6.94858084343562091580231792265, −6.58188660656238793184164059022, −4.92271026265499184264855244526, −4.35768998077195623278183191858, −3.37297314119773595030376885592, −1.92216567264704060052244492666, −0.20935810779684221209552993690,
1.83994628920741909558490762796, 3.07314379990765318279080540854, 4.09582377129762790925337993451, 5.16218299409769242545182470597, 6.01827787505478317548527868390, 6.93837171342014713745268461700, 8.213171684945024129473622952594, 8.313470031083564203581368263680, 9.496519304934470175352751868317, 10.25949087841005958000339057767
