L(s) = 1 | + (0.5 − 0.866i)5-s + (−0.173 − 0.984i)9-s + (−1.70 − 0.300i)13-s + (0.673 − 0.118i)17-s + (−0.499 − 0.866i)25-s + (−0.173 − 0.300i)29-s + (0.173 − 0.984i)37-s + (0.326 − 1.85i)41-s + (−0.939 − 0.342i)45-s + (−0.766 + 0.642i)49-s + (−0.592 + 1.62i)53-s + (−0.0603 + 0.342i)61-s + (−1.11 + 1.32i)65-s − 1.73i·73-s + (−0.939 + 0.342i)81-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)5-s + (−0.173 − 0.984i)9-s + (−1.70 − 0.300i)13-s + (0.673 − 0.118i)17-s + (−0.499 − 0.866i)25-s + (−0.173 − 0.300i)29-s + (0.173 − 0.984i)37-s + (0.326 − 1.85i)41-s + (−0.939 − 0.342i)45-s + (−0.766 + 0.642i)49-s + (−0.592 + 1.62i)53-s + (−0.0603 + 0.342i)61-s + (−1.11 + 1.32i)65-s − 1.73i·73-s + (−0.939 + 0.342i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.238 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.238 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.073762912\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.073762912\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 37 | \( 1 + (-0.173 + 0.984i)T \) |
good | 3 | \( 1 + (0.173 + 0.984i)T^{2} \) |
| 7 | \( 1 + (0.766 - 0.642i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (1.70 + 0.300i)T + (0.939 + 0.342i)T^{2} \) |
| 17 | \( 1 + (-0.673 + 0.118i)T + (0.939 - 0.342i)T^{2} \) |
| 19 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 41 | \( 1 + (-0.326 + 1.85i)T + (-0.939 - 0.342i)T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.592 - 1.62i)T + (-0.766 - 0.642i)T^{2} \) |
| 59 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 61 | \( 1 + (0.0603 - 0.342i)T + (-0.939 - 0.342i)T^{2} \) |
| 67 | \( 1 + (0.766 - 0.642i)T^{2} \) |
| 71 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 73 | \( 1 + 1.73iT - T^{2} \) |
| 79 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 83 | \( 1 + (-0.939 + 0.342i)T^{2} \) |
| 89 | \( 1 + (-1.76 - 0.642i)T + (0.766 + 0.642i)T^{2} \) |
| 97 | \( 1 + (-1.70 - 0.984i)T + (0.5 + 0.866i)T^{2} \) |
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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
−9.024803872625387357777787651301, −7.85504520248727368614404572025, −7.36612355848177678278158711701, −6.29592894752241670018291373551, −5.61350730012024056045731543384, −4.91699345180268470383801249049, −4.06177365460030625128401030788, −2.98023392957289680448694892082, −1.99479666487297060881955332557, −0.63792047485703077912084748771,
1.78280011909008805058367522047, 2.58389485632583811498182655909, 3.36006806512681040598811595529, 4.71392421655945255805320476196, 5.19740456099567340326347014194, 6.18013312781564511632797836558, 6.92289170968070159009751576550, 7.63450614276196363248057008342, 8.189318631846809404344429868787, 9.342970442838997783641519704969