L(s) = 1 | + (0.766 + 0.642i)2-s + (0.939 − 0.342i)3-s + (0.173 + 0.984i)4-s + (0.939 + 0.342i)6-s + (−0.500 + 0.866i)8-s + (0.5 − 0.866i)11-s + (0.499 + 0.866i)12-s + (−0.939 + 0.342i)16-s + (1.53 + 1.28i)17-s + (0.939 − 0.342i)22-s + (−0.173 + 0.984i)24-s + (−0.939 − 0.342i)25-s + (−0.499 + 0.866i)27-s + (−0.939 − 0.342i)32-s + (0.173 − 0.984i)33-s + (0.347 + 1.96i)34-s + ⋯ |
L(s) = 1 | + (0.766 + 0.642i)2-s + (0.939 − 0.342i)3-s + (0.173 + 0.984i)4-s + (0.939 + 0.342i)6-s + (−0.500 + 0.866i)8-s + (0.5 − 0.866i)11-s + (0.499 + 0.866i)12-s + (−0.939 + 0.342i)16-s + (1.53 + 1.28i)17-s + (0.939 − 0.342i)22-s + (−0.173 + 0.984i)24-s + (−0.939 − 0.342i)25-s + (−0.499 + 0.866i)27-s + (−0.939 − 0.342i)32-s + (0.173 − 0.984i)33-s + (0.347 + 1.96i)34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.499 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.499 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.556180093\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.556180093\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.766 - 0.642i)T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \) |
| 5 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 17 | \( 1 + (-1.53 - 1.28i)T + (0.173 + 0.984i)T^{2} \) |
| 23 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 29 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \) |
| 43 | \( 1 + (-0.347 + 1.96i)T + (-0.939 - 0.342i)T^{2} \) |
| 47 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 53 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 59 | \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \) |
| 61 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 67 | \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \) |
| 71 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 73 | \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \) |
| 79 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 83 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (1.87 + 0.684i)T + (0.766 + 0.642i)T^{2} \) |
| 97 | \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \) |
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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
−8.749090972671818670668008080098, −8.109179140670212054212320198698, −7.74027043131350287857904363801, −6.83432412733017173713425009291, −5.85170755722400125605240723778, −5.54610683070887019395640178291, −4.14482564202360871784907291852, −3.56087509032473436438334635633, −2.80295473625207870785760040799, −1.69878754120843026987059563009,
1.31871239397045894815729063763, 2.49061118893622838472850498626, 3.16788614294526589651645975774, 3.93525383357316514256716777302, 4.71052538859762984216868679588, 5.57277645709845277665282392038, 6.40558725268815130584000551280, 7.40135665204493632844512928932, 8.033632113141193869270454077821, 9.314214085365305114729469506272