L(s) = 1 | + (−0.309 − 0.951i)2-s + (0.809 + 0.587i)3-s + (−0.309 + 0.951i)5-s + (0.309 − 0.951i)6-s + (−0.809 − 0.587i)8-s + (0.309 + 0.951i)9-s + 0.999·10-s + (−0.809 + 0.587i)15-s + (−0.309 + 0.951i)16-s + (−0.309 + 0.951i)17-s + (0.809 − 0.587i)18-s + (1.61 + 1.17i)19-s + 23-s + (−0.309 − 0.951i)24-s + (−0.809 − 0.587i)25-s + ⋯ |
L(s) = 1 | + (−0.309 − 0.951i)2-s + (0.809 + 0.587i)3-s + (−0.309 + 0.951i)5-s + (0.309 − 0.951i)6-s + (−0.809 − 0.587i)8-s + (0.309 + 0.951i)9-s + 0.999·10-s + (−0.809 + 0.587i)15-s + (−0.309 + 0.951i)16-s + (−0.309 + 0.951i)17-s + (0.809 − 0.587i)18-s + (1.61 + 1.17i)19-s + 23-s + (−0.309 − 0.951i)24-s + (−0.809 − 0.587i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0475i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0475i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.286408350\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.286408350\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.809 - 0.587i)T \) |
| 5 | \( 1 + (0.309 - 0.951i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
| 7 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 17 | \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \) |
| 19 | \( 1 + (-1.61 - 1.17i)T + (0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (-0.618 + 1.90i)T + (-0.809 - 0.587i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.809 - 0.587i)T^{2} \) |
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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
−9.723158544704179617033127069398, −8.926180710036742323620384467248, −8.002941075387897585211406845951, −7.29782507571564731032645704120, −6.34024317220435949670000842600, −5.34322857122422211015162360185, −3.92907886031585784586874071663, −3.43402950917374759970870653909, −2.61913304103316682779795176387, −1.64284458568754997248037075199,
1.05624308260381677301086606562, 2.59480321699753974404392106378, 3.38572822331247359133482970495, 4.78859160882589497474531494373, 5.45766615921848701240381182303, 6.67202825062743666841741407838, 7.22637720127534518403419375533, 7.77552314895992499160446706714, 8.597535486294732005470020635409, 9.156224258539143912231152273345