L(s) = 1 | + (−0.955 + 0.294i)3-s + (−0.0747 + 0.997i)7-s + (0.826 − 0.563i)9-s + (−1.72 + 0.829i)13-s + (0.0747 − 0.129i)19-s + (−0.222 − 0.974i)21-s + (0.0747 + 0.997i)25-s + (−0.623 + 0.781i)27-s + (−0.988 − 1.71i)31-s + (−0.722 + 0.108i)37-s + (1.40 − 1.29i)39-s + (0.367 + 1.61i)43-s + (−0.988 − 0.149i)49-s + (−0.0332 + 0.145i)57-s + (−1.23 + 0.185i)61-s + ⋯ |
L(s) = 1 | + (−0.955 + 0.294i)3-s + (−0.0747 + 0.997i)7-s + (0.826 − 0.563i)9-s + (−1.72 + 0.829i)13-s + (0.0747 − 0.129i)19-s + (−0.222 − 0.974i)21-s + (0.0747 + 0.997i)25-s + (−0.623 + 0.781i)27-s + (−0.988 − 1.71i)31-s + (−0.722 + 0.108i)37-s + (1.40 − 1.29i)39-s + (0.367 + 1.61i)43-s + (−0.988 − 0.149i)49-s + (−0.0332 + 0.145i)57-s + (−1.23 + 0.185i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 - 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 - 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3797101136\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3797101136\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.955 - 0.294i)T \) |
| 7 | \( 1 + (0.0747 - 0.997i)T \) |
good | 5 | \( 1 + (-0.0747 - 0.997i)T^{2} \) |
| 11 | \( 1 + (-0.365 - 0.930i)T^{2} \) |
| 13 | \( 1 + (1.72 - 0.829i)T + (0.623 - 0.781i)T^{2} \) |
| 17 | \( 1 + (0.733 + 0.680i)T^{2} \) |
| 19 | \( 1 + (-0.0747 + 0.129i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.733 - 0.680i)T^{2} \) |
| 29 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 31 | \( 1 + (0.988 + 1.71i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.722 - 0.108i)T + (0.955 - 0.294i)T^{2} \) |
| 41 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 43 | \( 1 + (-0.367 - 1.61i)T + (-0.900 + 0.433i)T^{2} \) |
| 47 | \( 1 + (0.988 + 0.149i)T^{2} \) |
| 53 | \( 1 + (-0.955 - 0.294i)T^{2} \) |
| 59 | \( 1 + (-0.0747 + 0.997i)T^{2} \) |
| 61 | \( 1 + (1.23 - 0.185i)T + (0.955 - 0.294i)T^{2} \) |
| 67 | \( 1 + (0.733 + 1.26i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 73 | \( 1 + (-0.0546 - 0.728i)T + (-0.988 + 0.149i)T^{2} \) |
| 79 | \( 1 + (0.988 - 1.71i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 89 | \( 1 + (-0.365 + 0.930i)T^{2} \) |
| 97 | \( 1 + 1.80T + T^{2} \) |
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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
−9.451896518039867443396249770397, −9.121513870248068038386120371229, −7.79976277347213923217202161243, −7.13233031361313934843782933801, −6.30586741684958357426794949225, −5.51592871179024218802962877329, −4.90936474134328193087995218648, −4.08778116352667512484497614658, −2.81230088966037015923332730853, −1.75857494847566921037715745747,
0.27983509071363694380779464788, 1.69627916949802787486959036657, 2.98986270719323247350820099690, 4.17551429705754175015214479295, 4.96306676366488030612389119541, 5.58454471692110599267161301770, 6.65758619401310712665797066835, 7.26859523323356671718792601038, 7.71304547493690112168224651268, 8.821861480080620157749133040422