L(s) = 1 | + (−0.309 − 0.951i)2-s + (0.809 − 0.587i)3-s + (−0.809 + 0.587i)4-s + (−0.309 − 0.951i)5-s + (−0.809 − 0.587i)6-s − 1.61·7-s + (0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s + (−0.809 + 0.587i)10-s + (−0.190 − 0.587i)11-s + (−0.309 + 0.951i)12-s + (0.500 + 1.53i)14-s + (−0.809 − 0.587i)15-s + (0.309 − 0.951i)16-s − 18-s + ⋯ |
L(s) = 1 | + (−0.309 − 0.951i)2-s + (0.809 − 0.587i)3-s + (−0.809 + 0.587i)4-s + (−0.309 − 0.951i)5-s + (−0.809 − 0.587i)6-s − 1.61·7-s + (0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s + (−0.809 + 0.587i)10-s + (−0.190 − 0.587i)11-s + (−0.309 + 0.951i)12-s + (0.500 + 1.53i)14-s + (−0.809 − 0.587i)15-s + (0.309 − 0.951i)16-s − 18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7250359177\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7250359177\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.309 + 0.951i)T \) |
| 3 | \( 1 + (-0.809 + 0.587i)T \) |
| 5 | \( 1 + (0.309 + 0.951i)T \) |
good | 7 | \( 1 + 1.61T + T^{2} \) |
| 11 | \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 29 | \( 1 + (-1.61 + 1.17i)T + (0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 71 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.618 - 1.90i)T + (-0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 89 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 97 | \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \) |
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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
−10.14026000249862335144895784802, −9.693507600777819284632008893034, −8.669158990471229483665836356675, −8.337536421615635499183523052506, −7.14254561741020520523290292026, −6.01834080393432966584567233959, −4.46165948994787499370132581884, −3.43394796637693852198834991532, −2.62389902806177915964267520957, −0.890317674643063229284811383072,
2.70509528835814710740052894529, 3.67476107723092059868908245222, 4.73415586335395170227535938226, 6.16801099675516020279824416309, 6.86649025843642921970021235882, 7.66981501816775883330459380997, 8.620230155859547483875702367982, 9.580470897731809417451843315092, 10.07883650341338401472111044932, 10.69806784290767850885714840162