L(s) = 1 | + (1 + i)3-s + i·5-s + i·9-s + i·11-s + (−1 + i)15-s + (−1 − i)23-s − 25-s + 2i·31-s + (−1 + i)33-s + (1 + i)37-s − 45-s + (1 − i)47-s − i·49-s + (1 − i)53-s − 55-s + ⋯ |
L(s) = 1 | + (1 + i)3-s + i·5-s + i·9-s + i·11-s + (−1 + i)15-s + (−1 − i)23-s − 25-s + 2i·31-s + (−1 + i)33-s + (1 + i)37-s − 45-s + (1 − i)47-s − i·49-s + (1 − i)53-s − 55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.702118650\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.702118650\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 11 | \( 1 - iT \) |
good | 3 | \( 1 + (-1 - i)T + iT^{2} \) |
| 7 | \( 1 + iT^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (1 + i)T + iT^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 - 2iT - T^{2} \) |
| 37 | \( 1 + (-1 - i)T + iT^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (-1 + i)T - iT^{2} \) |
| 53 | \( 1 + (-1 + i)T - iT^{2} \) |
| 59 | \( 1 + 2T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + (1 - i)T - iT^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + 2iT - T^{2} \) |
| 97 | \( 1 + (-1 - i)T + iT^{2} \) |
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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
−8.959413830882432028919420140206, −8.426804945928555706539348257693, −7.57432548371909978402923575997, −6.87904312531357524354101443112, −6.12002223052367527014788395765, −4.93672759219392977097245953046, −4.26871364490257042912855529142, −3.49148663243138970128139299962, −2.75972577348477865730905137322, −1.95986582775280986915833989711,
0.884116654868285036603463869265, 1.87878290293833525195146886545, 2.74235442698613735742205764445, 3.75576818628344640257844640158, 4.52294321957145122556567156863, 5.89491416418119443838896801987, 5.98997272316989416539037392683, 7.51366561642598946347535082897, 7.71306184159860454152181016566, 8.373121607941518872698173430142