| L(s) = 1 | − i·2-s + (1.06 + 1.36i)3-s − 4-s + (1.36 − 1.06i)6-s + (2.62 − 0.294i)7-s + i·8-s + (−0.716 + 2.91i)9-s + 1.43i·11-s + (−1.06 − 1.36i)12-s + 4.73i·13-s + (−0.294 − 2.62i)14-s + 16-s − 2.59·17-s + (2.91 + 0.716i)18-s + 2.72i·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + (0.616 + 0.786i)3-s − 0.5·4-s + (0.556 − 0.436i)6-s + (0.993 − 0.111i)7-s + 0.353i·8-s + (−0.238 + 0.971i)9-s + 0.431i·11-s + (−0.308 − 0.393i)12-s + 1.31i·13-s + (−0.0786 − 0.702i)14-s + 0.250·16-s − 0.630·17-s + (0.686 + 0.168i)18-s + 0.625i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.924820143\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.924820143\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + (-1.06 - 1.36i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.62 + 0.294i)T \) |
| good | 11 | \( 1 - 1.43iT - 11T^{2} \) |
| 13 | \( 1 - 4.73iT - 13T^{2} \) |
| 17 | \( 1 + 2.59T + 17T^{2} \) |
| 19 | \( 1 - 2.72iT - 19T^{2} \) |
| 23 | \( 1 + 2.82iT - 23T^{2} \) |
| 29 | \( 1 + 2.82iT - 29T^{2} \) |
| 31 | \( 1 - 5.91iT - 31T^{2} \) |
| 37 | \( 1 - 2.39T + 37T^{2} \) |
| 41 | \( 1 - 11.1T + 41T^{2} \) |
| 43 | \( 1 + 0.432T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 + 5.69iT - 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 - 1.05iT - 61T^{2} \) |
| 67 | \( 1 + 9.82T + 67T^{2} \) |
| 71 | \( 1 - 13.2iT - 71T^{2} \) |
| 73 | \( 1 - 7.58iT - 73T^{2} \) |
| 79 | \( 1 - 16.5T + 79T^{2} \) |
| 83 | \( 1 - 12.9T + 83T^{2} \) |
| 89 | \( 1 - 3.90T + 89T^{2} \) |
| 97 | \( 1 - 14.0iT - 97T^{2} \) |
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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
−9.993570064414559937256400654788, −9.293353449106572837827695515651, −8.548479872082943499671864207332, −7.87133108926971494005766761994, −6.72963385196200263540211818029, −5.30897641515993745011258508813, −4.44729449562709950054066106524, −3.96801880132424776936951150896, −2.55278126988216747032770406450, −1.69784905881433644529121566763,
0.847913853050507608860301150241, 2.32184513858266542252848332757, 3.48053585786902135638185739096, 4.72632351752647312949141139447, 5.69666853991835714373464226627, 6.48050558993119664549762100580, 7.70722467298178883522667196942, 7.80806303939957560699263966881, 8.797464506338472034303421163099, 9.403430022753804696517949190009
