L(s) = 1 | + (−0.939 − 0.342i)5-s + (−0.984 + 0.173i)9-s + (0.984 + 0.173i)13-s + (0.118 + 0.673i)17-s + (0.766 + 0.642i)25-s + (1.10 + 0.296i)29-s + (0.173 − 0.984i)37-s + (0.673 + 0.118i)41-s + (0.984 + 0.173i)45-s + (0.642 + 0.766i)49-s + (0.469 − 0.218i)53-s + (1.34 − 0.939i)61-s + (−0.866 − 0.5i)65-s + (1.36 + 1.36i)73-s + (0.939 − 0.342i)81-s + ⋯ |
L(s) = 1 | + (−0.939 − 0.342i)5-s + (−0.984 + 0.173i)9-s + (0.984 + 0.173i)13-s + (0.118 + 0.673i)17-s + (0.766 + 0.642i)25-s + (1.10 + 0.296i)29-s + (0.173 − 0.984i)37-s + (0.673 + 0.118i)41-s + (0.984 + 0.173i)45-s + (0.642 + 0.766i)49-s + (0.469 − 0.218i)53-s + (1.34 − 0.939i)61-s + (−0.866 − 0.5i)65-s + (1.36 + 1.36i)73-s + (0.939 − 0.342i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.217i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.217i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9740492854\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9740492854\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.939 + 0.342i)T \) |
| 37 | \( 1 + (-0.173 + 0.984i)T \) |
good | 3 | \( 1 + (0.984 - 0.173i)T^{2} \) |
| 7 | \( 1 + (-0.642 - 0.766i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.984 - 0.173i)T + (0.939 + 0.342i)T^{2} \) |
| 17 | \( 1 + (-0.118 - 0.673i)T + (-0.939 + 0.342i)T^{2} \) |
| 19 | \( 1 + (0.984 - 0.173i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-1.10 - 0.296i)T + (0.866 + 0.5i)T^{2} \) |
| 31 | \( 1 - iT^{2} \) |
| 41 | \( 1 + (-0.673 - 0.118i)T + (0.939 + 0.342i)T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (-0.469 + 0.218i)T + (0.642 - 0.766i)T^{2} \) |
| 59 | \( 1 + (0.642 - 0.766i)T^{2} \) |
| 61 | \( 1 + (-1.34 + 0.939i)T + (0.342 - 0.939i)T^{2} \) |
| 67 | \( 1 + (0.642 + 0.766i)T^{2} \) |
| 71 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 73 | \( 1 + (-1.36 - 1.36i)T + iT^{2} \) |
| 79 | \( 1 + (-0.642 - 0.766i)T^{2} \) |
| 83 | \( 1 + (-0.342 - 0.939i)T^{2} \) |
| 89 | \( 1 + (0.766 - 0.357i)T + (0.642 - 0.766i)T^{2} \) |
| 97 | \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−8.649549042880166627461202220315, −8.379402158007383193385534885116, −7.60980863131119412655429196115, −6.67502455181599489604750323692, −5.88600092770903266985619119445, −5.11021915254215613047930090498, −4.11837834371221374156900199153, −3.50004131091887853840304646324, −2.45055100381118556080234964447, −0.994729876300136376777135210251,
0.819524518260965400182535061946, 2.54783586090650312848475393402, 3.27601664271092639459299585828, 4.07049949025508409957748414499, 5.02104395878714992829452478322, 5.93529312831385438708703892472, 6.64541657229260222870525569930, 7.43746804229344455219227677759, 8.332797391665756482620581533815, 8.596379503304978747415972658998