| 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
−15.59461919004984204952321323569, −14.21237496700354657276706814216, −13.15884627493339383429814612152, −12.21051665090741868031115105838, −10.68372419403733204059333291826, −8.986147440681671677679816455417, −8.208046929617629467128032615792, −7.09193503008472694559106097842, −5.27019738107696468255483049268, −4.22564061217201658866752193569,
3.03544157255568814852703229915, 4.29878395807902601847606510864, 6.52495137986341477590907454457, 8.063910698916515610767798864707, 9.401820822599483704682833492791, 10.67475820275141773886786578035, 11.54553781733229543264173850657, 12.36784286101715037183258870678, 13.73708928432188607873060131214, 14.84625762244638749011823645362
