L(s) = 1 | + (−0.602 − 0.798i)2-s + (0.172 − 1.85i)3-s + (−0.273 + 0.961i)4-s + (−1.58 + 0.981i)6-s + (0.932 − 0.361i)8-s + (−2.43 − 0.455i)9-s + (0.465 − 1.63i)11-s + (1.73 + 0.673i)12-s + (−0.850 − 0.526i)16-s + (−1.58 − 0.614i)17-s + (1.10 + 2.21i)18-s + (0.831 + 1.66i)19-s + (−1.58 + 0.614i)22-s + (−0.510 − 1.79i)24-s + (−0.273 − 0.961i)25-s + ⋯ |
L(s) = 1 | + (−0.602 − 0.798i)2-s + (0.172 − 1.85i)3-s + (−0.273 + 0.961i)4-s + (−1.58 + 0.981i)6-s + (0.932 − 0.361i)8-s + (−2.43 − 0.455i)9-s + (0.465 − 1.63i)11-s + (1.73 + 0.673i)12-s + (−0.850 − 0.526i)16-s + (−1.58 − 0.614i)17-s + (1.10 + 2.21i)18-s + (0.831 + 1.66i)19-s + (−1.58 + 0.614i)22-s + (−0.510 − 1.79i)24-s + (−0.273 − 0.961i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.925 - 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.925 - 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6688464635\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6688464635\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.602 + 0.798i)T \) |
| 137 | \( 1 + (-0.445 + 0.895i)T \) |
good | 3 | \( 1 + (-0.172 + 1.85i)T + (-0.982 - 0.183i)T^{2} \) |
| 5 | \( 1 + (0.273 + 0.961i)T^{2} \) |
| 7 | \( 1 + (-0.0922 - 0.995i)T^{2} \) |
| 11 | \( 1 + (-0.465 + 1.63i)T + (-0.850 - 0.526i)T^{2} \) |
| 13 | \( 1 + (-0.0922 - 0.995i)T^{2} \) |
| 17 | \( 1 + (1.58 + 0.614i)T + (0.739 + 0.673i)T^{2} \) |
| 19 | \( 1 + (-0.831 - 1.66i)T + (-0.602 + 0.798i)T^{2} \) |
| 23 | \( 1 + (-0.445 - 0.895i)T^{2} \) |
| 29 | \( 1 + (-0.445 - 0.895i)T^{2} \) |
| 31 | \( 1 + (0.602 + 0.798i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 0.184T + T^{2} \) |
| 43 | \( 1 + (0.757 + 1.52i)T + (-0.602 + 0.798i)T^{2} \) |
| 47 | \( 1 + (-0.932 - 0.361i)T^{2} \) |
| 53 | \( 1 + (0.602 - 0.798i)T^{2} \) |
| 59 | \( 1 + (-1.18 - 0.221i)T + (0.932 + 0.361i)T^{2} \) |
| 61 | \( 1 + (-0.932 + 0.361i)T^{2} \) |
| 67 | \( 1 + (-1.37 - 1.25i)T + (0.0922 + 0.995i)T^{2} \) |
| 71 | \( 1 + (0.850 - 0.526i)T^{2} \) |
| 73 | \( 1 + (-0.136 + 0.124i)T + (0.0922 - 0.995i)T^{2} \) |
| 79 | \( 1 + (0.982 - 0.183i)T^{2} \) |
| 83 | \( 1 + (-1.37 + 0.533i)T + (0.739 - 0.673i)T^{2} \) |
| 89 | \( 1 + (0.537 - 0.711i)T + (-0.273 - 0.961i)T^{2} \) |
| 97 | \( 1 + (-0.149 + 0.526i)T + (-0.850 - 0.526i)T^{2} \) |
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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
−9.387818597478701830976200125892, −8.493412029419153263297438646353, −8.229581750036554284688808853589, −7.23359093606245978119823547129, −6.50188828864163367825267580126, −5.60850151790006321229554102839, −3.83264539119007287827956391225, −2.84263900123823060117526063964, −1.88774110119263994760970423241, −0.73137256922889119150712694412,
2.24885912363550974204577617232, 3.80736754372184021537044635554, 4.76145714019397996971401239360, 5.04652240100537501613148625343, 6.39189317493975448903603153267, 7.19869116722620540753695516656, 8.336967047780307265669978794898, 9.179438609824371975681273870048, 9.492901236601751550358796216931, 10.15767087059313485982452918702