L(s) = 1 | − 3-s + 3.04·5-s + 9-s + 3.93·11-s + 4.88·13-s − 3.04·15-s − 5.34·17-s + 2.30·19-s − 7.93·23-s + 4.25·25-s − 27-s + 5.55·29-s + 0.645·31-s − 3.93·33-s + 5.65·37-s − 4.88·39-s + 10.0·41-s + 8.91·43-s + 3.04·45-s + 6.61·47-s + 5.34·51-s + 1.25·53-s + 11.9·55-s − 2.30·57-s − 3.04·59-s − 2.97·61-s + 14.8·65-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.36·5-s + 0.333·9-s + 1.18·11-s + 1.35·13-s − 0.785·15-s − 1.29·17-s + 0.528·19-s − 1.65·23-s + 0.851·25-s − 0.192·27-s + 1.03·29-s + 0.115·31-s − 0.684·33-s + 0.929·37-s − 0.782·39-s + 1.57·41-s + 1.35·43-s + 0.453·45-s + 0.964·47-s + 0.748·51-s + 0.172·53-s + 1.61·55-s − 0.304·57-s − 0.396·59-s − 0.380·61-s + 1.84·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.477303546\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.477303546\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 3.04T + 5T^{2} \) |
| 11 | \( 1 - 3.93T + 11T^{2} \) |
| 13 | \( 1 - 4.88T + 13T^{2} \) |
| 17 | \( 1 + 5.34T + 17T^{2} \) |
| 19 | \( 1 - 2.30T + 19T^{2} \) |
| 23 | \( 1 + 7.93T + 23T^{2} \) |
| 29 | \( 1 - 5.55T + 29T^{2} \) |
| 31 | \( 1 - 0.645T + 31T^{2} \) |
| 37 | \( 1 - 5.65T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 - 8.91T + 43T^{2} \) |
| 47 | \( 1 - 6.61T + 47T^{2} \) |
| 53 | \( 1 - 1.25T + 53T^{2} \) |
| 59 | \( 1 + 3.04T + 59T^{2} \) |
| 61 | \( 1 + 2.97T + 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 - 4.67T + 73T^{2} \) |
| 79 | \( 1 + 1.05T + 79T^{2} \) |
| 83 | \( 1 + 8.60T + 83T^{2} \) |
| 89 | \( 1 - 4.85T + 89T^{2} \) |
| 97 | \( 1 - 18.3T + 97T^{2} \) |
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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
−8.457224094076325851054902429390, −7.44843924363016386939490434602, −6.51232843735151971634426323815, −6.01546601003918001181132913825, −5.80723571886015257991531232081, −4.45270962744138871239567962757, −4.04686055999708564411203983664, −2.70978645889823521791844248293, −1.77199919994115836227266757886, −0.969579294099863663035776260396,
0.969579294099863663035776260396, 1.77199919994115836227266757886, 2.70978645889823521791844248293, 4.04686055999708564411203983664, 4.45270962744138871239567962757, 5.80723571886015257991531232081, 6.01546601003918001181132913825, 6.51232843735151971634426323815, 7.44843924363016386939490434602, 8.457224094076325851054902429390