L(s) = 1 | + (0.707 − 0.707i)2-s − 1.00i·4-s + (0.707 + 0.707i)7-s + (−0.707 − 0.707i)8-s − 0.585i·11-s + (−0.0249 + 0.0249i)13-s + 1.00·14-s − 1.00·16-s + (−1.92 + 1.92i)17-s + 4.87i·19-s + (−0.414 − 0.414i)22-s + (5.15 + 5.15i)23-s + 0.0352i·26-s + (0.707 − 0.707i)28-s − 4.58·29-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s − 0.500i·4-s + (0.267 + 0.267i)7-s + (−0.250 − 0.250i)8-s − 0.176i·11-s + (−0.00691 + 0.00691i)13-s + 0.267·14-s − 0.250·16-s + (−0.466 + 0.466i)17-s + 1.11i·19-s + (−0.0883 − 0.0883i)22-s + (1.07 + 1.07i)23-s + 0.00691i·26-s + (0.133 − 0.133i)28-s − 0.850·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0387i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.378960570\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.378960570\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
good | 11 | \( 1 + 0.585iT - 11T^{2} \) |
| 13 | \( 1 + (0.0249 - 0.0249i)T - 13iT^{2} \) |
| 17 | \( 1 + (1.92 - 1.92i)T - 17iT^{2} \) |
| 19 | \( 1 - 4.87iT - 19T^{2} \) |
| 23 | \( 1 + (-5.15 - 5.15i)T + 23iT^{2} \) |
| 29 | \( 1 + 4.58T + 29T^{2} \) |
| 31 | \( 1 - 9.83T + 31T^{2} \) |
| 37 | \( 1 + (-2.87 - 2.87i)T + 37iT^{2} \) |
| 41 | \( 1 + 0.979iT - 41T^{2} \) |
| 43 | \( 1 + (4.27 - 4.27i)T - 43iT^{2} \) |
| 47 | \( 1 + (-6.32 + 6.32i)T - 47iT^{2} \) |
| 53 | \( 1 + (-1.56 - 1.56i)T + 53iT^{2} \) |
| 59 | \( 1 + 0.670T + 59T^{2} \) |
| 61 | \( 1 - 2.35T + 61T^{2} \) |
| 67 | \( 1 + (-5.08 - 5.08i)T + 67iT^{2} \) |
| 71 | \( 1 - 2.92iT - 71T^{2} \) |
| 73 | \( 1 + (-2.51 + 2.51i)T - 73iT^{2} \) |
| 79 | \( 1 + 8.71iT - 79T^{2} \) |
| 83 | \( 1 + (-2.99 - 2.99i)T + 83iT^{2} \) |
| 89 | \( 1 + 9.72T + 89T^{2} \) |
| 97 | \( 1 + (-4.74 - 4.74i)T + 97iT^{2} \) |
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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
−8.694130252224563941425830107783, −8.035648219806644287762528126600, −7.12241261811959331332349425814, −6.23228333437830771452084119651, −5.57499260095062393533049776711, −4.78675378995439360040761893574, −3.92232654573701573645484696858, −3.12464286353060354876683231621, −2.11454345679736615255568594777, −1.12870159742794260796439844728,
0.70406815817534533691427932371, 2.29985945266253156283004579131, 3.08089170336717241536480971263, 4.28430726169057202800501365329, 4.73952972846670617191514672312, 5.56368099952998296976804729293, 6.59887149628713875685291074480, 6.99173231641700821070476058214, 7.82236451143797545870380711029, 8.610905833843468011879839371667