L(s) = 1 | + (−0.809 + 0.587i)3-s + (0.587 + 1.80i)5-s + (0.809 + 0.587i)7-s + (0.309 − 0.951i)9-s + (−0.951 − 0.309i)11-s + (−1.53 − 1.11i)15-s + (−0.363 − 1.11i)17-s + (−1.30 + 0.951i)19-s − 21-s − 1.17·23-s + (−2.11 + 1.53i)25-s + (0.309 + 0.951i)27-s + (0.190 − 0.587i)31-s + (0.951 − 0.309i)33-s + (−0.587 + 1.80i)35-s + ⋯ |
L(s) = 1 | + (−0.809 + 0.587i)3-s + (0.587 + 1.80i)5-s + (0.809 + 0.587i)7-s + (0.309 − 0.951i)9-s + (−0.951 − 0.309i)11-s + (−1.53 − 1.11i)15-s + (−0.363 − 1.11i)17-s + (−1.30 + 0.951i)19-s − 21-s − 1.17·23-s + (−2.11 + 1.53i)25-s + (0.309 + 0.951i)27-s + (0.190 − 0.587i)31-s + (0.951 − 0.309i)33-s + (−0.587 + 1.80i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0457i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0457i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7036408695\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7036408695\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.809 - 0.587i)T \) |
| 7 | \( 1 + (-0.809 - 0.587i)T \) |
| 11 | \( 1 + (0.951 + 0.309i)T \) |
good | 5 | \( 1 + (-0.587 - 1.80i)T + (-0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (0.363 + 1.11i)T + (-0.809 + 0.587i)T^{2} \) |
| 19 | \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + 1.17T + T^{2} \) |
| 29 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (1.53 - 1.11i)T + (0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 79 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 89 | \( 1 - 1.17T + T^{2} \) |
| 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
−9.287398369063533057792547012617, −8.215303358112915695139728293710, −7.57926141659690302308864211995, −6.55258001361832325124254038615, −6.16031699186649924073454097122, −5.46104257266431162659109093004, −4.65114392014869321237881556498, −3.63111398919004348240314131366, −2.68897986802860455248264547355, −1.96804346839380412015027733335,
0.41447060925605578660689987229, 1.70349140780834837541879332727, 2.08911103483758923107725939196, 4.18496096172903575975185363510, 4.63248740979043947257784881440, 5.29876425752043288945529401029, 5.92891660339658696212195091162, 6.78632135875767263902096208440, 7.73812403924729053493334893378, 8.376376919495140495854071429055