L(s) = 1 | + (−0.707 + 0.707i)2-s + (−0.382 − 0.923i)3-s − 1.00i·4-s + (0.382 + 0.923i)5-s + (0.923 + 0.382i)6-s + (0.707 + 0.707i)7-s + (0.707 + 0.707i)8-s + (−0.707 + 0.707i)9-s + (−0.923 − 0.382i)10-s + (−0.923 + 0.382i)12-s + (−1.30 + 1.30i)13-s − 1.00·14-s + (0.707 − 0.707i)15-s − 1.00·16-s − i·18-s − 0.765i·19-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)2-s + (−0.382 − 0.923i)3-s − 1.00i·4-s + (0.382 + 0.923i)5-s + (0.923 + 0.382i)6-s + (0.707 + 0.707i)7-s + (0.707 + 0.707i)8-s + (−0.707 + 0.707i)9-s + (−0.923 − 0.382i)10-s + (−0.923 + 0.382i)12-s + (−1.30 + 1.30i)13-s − 1.00·14-s + (0.707 − 0.707i)15-s − 1.00·16-s − i·18-s − 0.765i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6243867529\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6243867529\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 3 | \( 1 + (0.382 + 0.923i)T \) |
| 5 | \( 1 + (-0.382 - 0.923i)T \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
good | 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (1.30 - 1.30i)T - iT^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 + 0.765iT - T^{2} \) |
| 23 | \( 1 + (-1 - i)T + iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 - 0.765T + T^{2} \) |
| 61 | \( 1 - 1.84T + T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + 1.41iT - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + 1.41iT - T^{2} \) |
| 83 | \( 1 + (0.541 + 0.541i)T + iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−10.59653593808114949687541551226, −9.520779880932448952371768638761, −8.893554792696618264178820115601, −7.76829920204580428348066715432, −7.12224533139881012306149312622, −6.57440665243773297820885064375, −5.54797725853779909750254374669, −4.84550912242740189153913146532, −2.55766681738740449173356675052, −1.75471889459808690040085823476,
0.872934332713651999777288495447, 2.55395093085680873567685738690, 3.88125212930056334834536621952, 4.77378272014413697342364915032, 5.45423428445852757565733926697, 6.99790430901327671843982117814, 8.095978429990034181111507114499, 8.618633601865618886710011815077, 9.733785380786163283069047174161, 10.10948903830152119357761729468