L(s) = 1 | + (1.11 + 0.872i)2-s + i·3-s + (0.477 + 1.94i)4-s + 0.970i·5-s + (−0.872 + 1.11i)6-s − 4.31·7-s + (−1.16 + 2.57i)8-s − 9-s + (−0.846 + 1.07i)10-s + 3.90·11-s + (−1.94 + 0.477i)12-s + 1.84·13-s + (−4.80 − 3.76i)14-s − 0.970·15-s + (−3.54 + 1.85i)16-s + 0.465i·17-s + ⋯ |
L(s) = 1 | + (0.786 + 0.617i)2-s + 0.577i·3-s + (0.238 + 0.971i)4-s + 0.433i·5-s + (−0.356 + 0.454i)6-s − 1.63·7-s + (−0.411 + 0.911i)8-s − 0.333·9-s + (−0.267 + 0.341i)10-s + 1.17·11-s + (−0.560 + 0.137i)12-s + 0.511·13-s + (−1.28 − 1.00i)14-s − 0.250·15-s + (−0.886 + 0.463i)16-s + 0.112i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.559 - 0.829i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.559 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.800434 + 1.50504i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.800434 + 1.50504i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.11 - 0.872i)T \) |
| 3 | \( 1 - iT \) |
| 23 | \( 1 + (1.65 + 4.50i)T \) |
good | 5 | \( 1 - 0.970iT - 5T^{2} \) |
| 7 | \( 1 + 4.31T + 7T^{2} \) |
| 11 | \( 1 - 3.90T + 11T^{2} \) |
| 13 | \( 1 - 1.84T + 13T^{2} \) |
| 17 | \( 1 - 0.465iT - 17T^{2} \) |
| 19 | \( 1 - 6.89T + 19T^{2} \) |
| 29 | \( 1 - 1.41T + 29T^{2} \) |
| 31 | \( 1 - 2.44iT - 31T^{2} \) |
| 37 | \( 1 - 11.3iT - 37T^{2} \) |
| 41 | \( 1 + 2.17T + 41T^{2} \) |
| 43 | \( 1 + 5.37T + 43T^{2} \) |
| 47 | \( 1 + 8.65iT - 47T^{2} \) |
| 53 | \( 1 + 10.2iT - 53T^{2} \) |
| 59 | \( 1 + 8.30iT - 59T^{2} \) |
| 61 | \( 1 + 7.91iT - 61T^{2} \) |
| 67 | \( 1 + 2.57T + 67T^{2} \) |
| 71 | \( 1 - 6.53iT - 71T^{2} \) |
| 73 | \( 1 + 1.48T + 73T^{2} \) |
| 79 | \( 1 - 12.2T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 + 4.10iT - 89T^{2} \) |
| 97 | \( 1 + 5.32iT - 97T^{2} \) |
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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
−12.25189418240954025007774398548, −11.52767205135921124945622162820, −10.23532193324670406240505611435, −9.364236218046823958378506803338, −8.347928885308128302869846500366, −6.68233388267889202794053693880, −6.53163114597021732821061553865, −5.12236256789024623995045015129, −3.69092053601108870987984726348, −3.10738923340747471845936544370,
1.14358032784187294768445261479, 2.99888859377750675742045485584, 3.93891319208023164487318892013, 5.56957113059338470736127711760, 6.37228956190205816566200222196, 7.29777452809949871259364708887, 9.164555858549515166975034220276, 9.538149162358269708992460510823, 10.83935872685427164886369713703, 11.98942481816015268624611851073