| L(s) = 1 | − 5.19·5-s + 5.19·7-s − 3·11-s + 10.3i·13-s − 6i·17-s + 2i·19-s − 10.3i·23-s + 2·25-s − 20.7·29-s + 36.3·31-s − 27·35-s + 51.9i·37-s − 42i·41-s + 4i·43-s − 41.5i·47-s + ⋯ |
| L(s) = 1 | − 1.03·5-s + 0.742·7-s − 0.272·11-s + 0.799i·13-s − 0.352i·17-s + 0.105i·19-s − 0.451i·23-s + 0.0800·25-s − 0.716·29-s + 1.17·31-s − 0.771·35-s + 1.40i·37-s − 1.02i·41-s + 0.0930i·43-s − 0.884i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5092219001\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5092219001\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 5.19T + 25T^{2} \) |
| 7 | \( 1 - 5.19T + 49T^{2} \) |
| 11 | \( 1 + 3T + 121T^{2} \) |
| 13 | \( 1 - 10.3iT - 169T^{2} \) |
| 17 | \( 1 + 6iT - 289T^{2} \) |
| 19 | \( 1 - 2iT - 361T^{2} \) |
| 23 | \( 1 + 10.3iT - 529T^{2} \) |
| 29 | \( 1 + 20.7T + 841T^{2} \) |
| 31 | \( 1 - 36.3T + 961T^{2} \) |
| 37 | \( 1 - 51.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 42iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 4iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 41.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 67.5T + 2.80e3T^{2} \) |
| 59 | \( 1 + 66T + 3.48e3T^{2} \) |
| 61 | \( 1 + 62.3iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 44iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 135. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 29T + 5.32e3T^{2} \) |
| 79 | \( 1 + 83.1T + 6.24e3T^{2} \) |
| 83 | \( 1 + 99T + 6.88e3T^{2} \) |
| 89 | \( 1 + 144iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 31T + 9.40e3T^{2} \) |
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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
−8.592114644824774761902174754377, −8.132164134219434437835504544047, −7.32444763902632325328836337306, −6.61303001271759251238341206376, −5.44498057088042011747058598944, −4.56988779334780610860884075352, −3.94240681115773580642515274321, −2.79189090336177341468061300931, −1.57370922179969173008880038014, −0.14697434540065010505018115250,
1.17711701584061049828047003537, 2.55910074868419097011774086139, 3.62728311991604845822102503949, 4.41817628224148559864046590803, 5.29354065929969756499383189974, 6.16832582277294264196511238090, 7.37913972644824208467373050731, 7.82931461316532227832077243035, 8.423263720118836182280152368731, 9.357690221991200576583930541407
