L(s) = 1 | − i·2-s + (0.707 + 0.707i)3-s − 4-s + (0.707 − 0.707i)6-s + (−0.707 − 0.707i)7-s + i·8-s + 11-s + (−0.707 − 0.707i)12-s + (−0.707 − 0.707i)13-s + (−0.707 + 0.707i)14-s + 16-s − 1.41i·19-s − 1.00i·21-s − i·22-s − i·23-s + (−0.707 + 0.707i)24-s + ⋯ |
L(s) = 1 | − i·2-s + (0.707 + 0.707i)3-s − 4-s + (0.707 − 0.707i)6-s + (−0.707 − 0.707i)7-s + i·8-s + 11-s + (−0.707 − 0.707i)12-s + (−0.707 − 0.707i)13-s + (−0.707 + 0.707i)14-s + 16-s − 1.41i·19-s − 1.00i·21-s − i·22-s − i·23-s + (−0.707 + 0.707i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0985 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0985 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.168033446\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.168033446\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 7 | \( 1 + (0.707 + 0.707i)T \) |
| 13 | \( 1 + (0.707 + 0.707i)T \) |
good | 3 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 11 | \( 1 - T + T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + 1.41iT - T^{2} \) |
| 23 | \( 1 + iT - T^{2} \) |
| 29 | \( 1 + (-1 - i)T + iT^{2} \) |
| 31 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 37 | \( 1 + iT - T^{2} \) |
| 41 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 43 | \( 1 + (1 - i)T - iT^{2} \) |
| 47 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( 1 + (1 + i)T + iT^{2} \) |
| 73 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 79 | \( 1 - iT - T^{2} \) |
| 83 | \( 1 + 1.41iT - T^{2} \) |
| 89 | \( 1 - iT^{2} \) |
| 97 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
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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
−9.540880618589220159740423536692, −9.025185461229121249460919181051, −8.365639281395802407931887925178, −7.12923469345733971138753500651, −6.29868285608613215594744481905, −4.70650669453271665325662552403, −4.40881658431381794246909072262, −3.04664880330489875182169888582, −2.96678280954904086361030421509, −0.990448338193624039235069371675,
1.63655030766313258869036750528, 2.94575877607889008449941534544, 3.99005742899899529720189979456, 5.06203576954810935966537068903, 6.11880276767701033195038976551, 6.68417769564901880737495430120, 7.45800697668475361565124174411, 8.259683147057823956147059075477, 8.865756647866876781372466849419, 9.602969250482093923742990212749