L(s) = 1 | + (0.826 + 0.563i)4-s + (−1.63 + 0.246i)5-s + (−0.988 + 0.149i)7-s + (−0.733 − 0.680i)9-s + (−1.40 + 1.29i)11-s + (0.365 + 0.930i)16-s + (−0.134 − 1.79i)17-s + (−0.5 + 0.866i)19-s + (−1.48 − 0.716i)20-s + (−0.109 + 1.46i)23-s + (1.65 − 0.510i)25-s + (−0.900 − 0.433i)28-s + (1.57 − 0.487i)35-s + (−0.222 − 0.974i)36-s + (0.0931 − 0.116i)43-s + (−1.88 + 0.284i)44-s + ⋯ |
L(s) = 1 | + (0.826 + 0.563i)4-s + (−1.63 + 0.246i)5-s + (−0.988 + 0.149i)7-s + (−0.733 − 0.680i)9-s + (−1.40 + 1.29i)11-s + (0.365 + 0.930i)16-s + (−0.134 − 1.79i)17-s + (−0.5 + 0.866i)19-s + (−1.48 − 0.716i)20-s + (−0.109 + 1.46i)23-s + (1.65 − 0.510i)25-s + (−0.900 − 0.433i)28-s + (1.57 − 0.487i)35-s + (−0.222 − 0.974i)36-s + (0.0931 − 0.116i)43-s + (−1.88 + 0.284i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.918 - 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.918 - 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3270829405\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3270829405\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (0.988 - 0.149i)T \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
good | 2 | \( 1 + (-0.826 - 0.563i)T^{2} \) |
| 3 | \( 1 + (0.733 + 0.680i)T^{2} \) |
| 5 | \( 1 + (1.63 - 0.246i)T + (0.955 - 0.294i)T^{2} \) |
| 11 | \( 1 + (1.40 - 1.29i)T + (0.0747 - 0.997i)T^{2} \) |
| 13 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 17 | \( 1 + (0.134 + 1.79i)T + (-0.988 + 0.149i)T^{2} \) |
| 23 | \( 1 + (0.109 - 1.46i)T + (-0.988 - 0.149i)T^{2} \) |
| 29 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.365 + 0.930i)T^{2} \) |
| 41 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 43 | \( 1 + (-0.0931 + 0.116i)T + (-0.222 - 0.974i)T^{2} \) |
| 47 | \( 1 + (-0.698 - 0.215i)T + (0.826 + 0.563i)T^{2} \) |
| 53 | \( 1 + (-0.365 - 0.930i)T^{2} \) |
| 59 | \( 1 + (-0.955 - 0.294i)T^{2} \) |
| 61 | \( 1 + (1.21 - 0.825i)T + (0.365 - 0.930i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 73 | \( 1 + (0.955 - 0.294i)T + (0.826 - 0.563i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.0332 - 0.145i)T + (-0.900 - 0.433i)T^{2} \) |
| 89 | \( 1 + (-0.0747 - 0.997i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.77466600464092741885386467000, −9.857537287753525853277624122604, −8.875058697475451043159324591934, −7.71895924374639848301986100956, −7.46560372425716084302212046136, −6.66383809408964407547332863656, −5.48003308635139232281967532420, −4.15221895815417283614898126943, −3.22903713977861968459193595555, −2.60488781548703194351160334150,
0.28453940898416654397080149977, 2.56166807712821985441792158982, 3.34094963509337701320773502815, 4.54416361365852207518825872182, 5.71337068434517124819701419346, 6.41387630304121533829432492091, 7.47491059180326357423503728637, 8.248157978388322175791794536417, 8.725026972727036140748664661591, 10.36964802931660956946299724660