| 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
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9581739384\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9581739384\) |
| \(L(2)\) |
|
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
−9.605429125665076892598798470164, −8.769322295254852722885599500131, −7.920148878724806315261290306436, −6.71847499070511113570951967461, −6.38756003435055793623449285546, −5.44976131526455953220479871801, −4.55762887484119338162372333346, −3.41794431113599411352575589463, −2.43277052180419404669013920859, −1.43705176976939174418436352829,
0.23438622214939444361598676802, 1.69837230091309245635968503785, 2.74794139697454009294743285757, 3.59438287703672873828980107757, 4.88875746991812594103161199022, 5.72863729608411940676448386197, 6.23429026901629764606566905380, 7.26122030286324217510299889284, 7.975655930435271353735162016871, 9.218006167754270715504542391994
