L(s) = 1 | − i·2-s − 4-s + (−1 + i)5-s + i·8-s − i·9-s + (1 + i)10-s + 16-s − 17-s − 18-s + (1 − i)20-s − i·25-s + (1 − i)29-s − i·32-s + i·34-s + i·36-s + (−1 + i)37-s + ⋯ |
L(s) = 1 | − i·2-s − 4-s + (−1 + i)5-s + i·8-s − i·9-s + (1 + i)10-s + 16-s − 17-s − 18-s + (1 − i)20-s − i·25-s + (1 − i)29-s − i·32-s + i·34-s + i·36-s + (−1 + i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4210945963\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4210945963\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 17 | \( 1 + T \) |
good | 3 | \( 1 + iT^{2} \) |
| 5 | \( 1 + (1 - i)T - iT^{2} \) |
| 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 - iT^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 + (-1 + i)T - iT^{2} \) |
| 31 | \( 1 + iT^{2} \) |
| 37 | \( 1 + (1 - i)T - iT^{2} \) |
| 41 | \( 1 + (-1 - i)T + iT^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 + (1 + i)T + iT^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + iT^{2} \) |
| 73 | \( 1 + (-1 + i)T - iT^{2} \) |
| 79 | \( 1 - iT^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (1 - i)T - iT^{2} \) |
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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
−14.84625762244638749011823645362, −13.73708928432188607873060131214, −12.36784286101715037183258870678, −11.54553781733229543264173850657, −10.67475820275141773886786578035, −9.401820822599483704682833492791, −8.063910698916515610767798864707, −6.52495137986341477590907454457, −4.29878395807902601847606510864, −3.03544157255568814852703229915,
4.22564061217201658866752193569, 5.27019738107696468255483049268, 7.09193503008472694559106097842, 8.208046929617629467128032615792, 8.986147440681671677679816455417, 10.68372419403733204059333291826, 12.21051665090741868031115105838, 13.15884627493339383429814612152, 14.21237496700354657276706814216, 15.59461919004984204952321323569