L(s) = 1 | + (0.654 − 0.755i)2-s + (−0.142 − 0.989i)4-s + (0.959 + 0.281i)7-s + (−0.841 − 0.540i)8-s + (0.415 + 0.909i)9-s + (0.817 + 0.708i)11-s + (0.841 − 0.540i)14-s + (−0.959 + 0.281i)16-s + (0.959 + 0.281i)18-s + (1.07 − 0.153i)22-s + (0.415 − 0.909i)23-s + (−0.654 − 0.755i)25-s + (0.142 − 0.989i)28-s + (−1.49 + 0.215i)29-s + (−0.415 + 0.909i)32-s + ⋯ |
L(s) = 1 | + (0.654 − 0.755i)2-s + (−0.142 − 0.989i)4-s + (0.959 + 0.281i)7-s + (−0.841 − 0.540i)8-s + (0.415 + 0.909i)9-s + (0.817 + 0.708i)11-s + (0.841 − 0.540i)14-s + (−0.959 + 0.281i)16-s + (0.959 + 0.281i)18-s + (1.07 − 0.153i)22-s + (0.415 − 0.909i)23-s + (−0.654 − 0.755i)25-s + (0.142 − 0.989i)28-s + (−1.49 + 0.215i)29-s + (−0.415 + 0.909i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.561 + 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.561 + 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.642366482\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.642366482\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.654 + 0.755i)T \) |
| 7 | \( 1 + (-0.959 - 0.281i)T \) |
| 23 | \( 1 + (-0.415 + 0.909i)T \) |
good | 3 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 5 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 11 | \( 1 + (-0.817 - 0.708i)T + (0.142 + 0.989i)T^{2} \) |
| 13 | \( 1 + (0.841 - 0.540i)T^{2} \) |
| 17 | \( 1 + (-0.959 + 0.281i)T^{2} \) |
| 19 | \( 1 + (-0.959 - 0.281i)T^{2} \) |
| 29 | \( 1 + (1.49 - 0.215i)T + (0.959 - 0.281i)T^{2} \) |
| 31 | \( 1 + (0.415 - 0.909i)T^{2} \) |
| 37 | \( 1 + (0.797 + 1.74i)T + (-0.654 + 0.755i)T^{2} \) |
| 41 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 43 | \( 1 + (0.983 - 1.53i)T + (-0.415 - 0.909i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (1.25 + 0.368i)T + (0.841 + 0.540i)T^{2} \) |
| 59 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 61 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 67 | \( 1 + (-1.49 + 1.29i)T + (0.142 - 0.989i)T^{2} \) |
| 71 | \( 1 + (0.817 - 0.708i)T + (0.142 - 0.989i)T^{2} \) |
| 73 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 79 | \( 1 + (-1.25 + 0.368i)T + (0.841 - 0.540i)T^{2} \) |
| 83 | \( 1 + (-0.654 + 0.755i)T^{2} \) |
| 89 | \( 1 + (0.415 + 0.909i)T^{2} \) |
| 97 | \( 1 + (-0.654 - 0.755i)T^{2} \) |
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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
−9.844601984495049674204754468687, −9.149439000844035750940565669562, −8.156481796603167141950631639380, −7.23568481976786014251866630156, −6.24611427844938903523811249974, −5.18925244092459453636521621180, −4.60958789062023474177007075638, −3.76364664476305600876595432782, −2.28264260212165077178205197842, −1.63853294374003482887839272950,
1.59148128789563529631093880401, 3.40814194761809712698079516969, 3.90810214861054331266521366863, 5.01546652646316360446289503357, 5.78415127725771112844561681324, 6.72687460907467816735765836820, 7.36612460307308778230134501352, 8.248174571209902162913392818127, 8.998476987253118759688265079606, 9.740770341571299788925945773963