L(s) = 1 | + (0.766 + 0.642i)2-s + (0.173 + 0.984i)4-s + (−1.43 − 0.524i)5-s + (−0.500 + 0.866i)8-s + (−0.766 − 1.32i)10-s + (−1.43 + 1.20i)13-s + (−0.939 + 0.342i)16-s + (−0.173 − 0.300i)17-s + (0.266 − 1.50i)20-s + (1.03 + 0.866i)25-s − 1.87·26-s + (−1.43 − 1.20i)29-s + (−0.939 − 0.342i)32-s + (0.0603 − 0.342i)34-s + (−0.173 − 0.300i)37-s + ⋯ |
L(s) = 1 | + (0.766 + 0.642i)2-s + (0.173 + 0.984i)4-s + (−1.43 − 0.524i)5-s + (−0.500 + 0.866i)8-s + (−0.766 − 1.32i)10-s + (−1.43 + 1.20i)13-s + (−0.939 + 0.342i)16-s + (−0.173 − 0.300i)17-s + (0.266 − 1.50i)20-s + (1.03 + 0.866i)25-s − 1.87·26-s + (−1.43 − 1.20i)29-s + (−0.939 − 0.342i)32-s + (0.0603 − 0.342i)34-s + (−0.173 − 0.300i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.893 + 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.893 + 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3718630285\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3718630285\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.766 - 0.642i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2} \) |
| 7 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 11 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 13 | \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \) |
| 17 | \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 29 | \( 1 + (1.43 + 1.20i)T + (0.173 + 0.984i)T^{2} \) |
| 31 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 37 | \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \) |
| 43 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 47 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 61 | \( 1 + (0.326 - 1.85i)T + (-0.939 - 0.342i)T^{2} \) |
| 67 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 71 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 83 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 89 | \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \) |
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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
−9.117283722847167701084217718768, −8.422176438718881779750083839873, −7.57423620080739032860372010519, −7.30489372548141442246505552352, −6.43220677745399886124302762030, −5.36173074199659232432722646816, −4.55547602048486235875568026375, −4.19209665795467172706359506403, −3.24309694845951646606048330423, −2.10753778304624643703397961642,
0.16581360023299951085227800625, 1.93534425914201360055297347403, 3.19713960841083029383661701739, 3.41853163382407628374182716134, 4.58182359133619191574023018670, 5.11526750189451234401586849210, 6.11983208193349716484148117014, 7.14158072318625019075288474723, 7.53384837222989123705412919797, 8.428782053939428380581642918139