| L(s) = 1 | + 0.792i·2-s + (1.18 + 1.26i)3-s + 1.37·4-s + (−1 + 0.939i)6-s + (2 + 1.73i)7-s + 2.67i·8-s + (−0.186 + 2.99i)9-s − 2.52i·11-s + (1.62 + 1.73i)12-s − 4.10i·13-s + (−1.37 + 1.58i)14-s + 0.627·16-s − 4.37·17-s + (−2.37 − 0.147i)18-s + 3.46i·19-s + ⋯ |
| L(s) = 1 | + 0.560i·2-s + (0.684 + 0.728i)3-s + 0.686·4-s + (−0.408 + 0.383i)6-s + (0.755 + 0.654i)7-s + 0.944i·8-s + (−0.0620 + 0.998i)9-s − 0.761i·11-s + (0.469 + 0.499i)12-s − 1.13i·13-s + (−0.366 + 0.423i)14-s + 0.156·16-s − 1.06·17-s + (−0.559 − 0.0347i)18-s + 0.794i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0406 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0406 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.54774 + 1.61194i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.54774 + 1.61194i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.18 - 1.26i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2 - 1.73i)T \) |
| good | 2 | \( 1 - 0.792iT - 2T^{2} \) |
| 11 | \( 1 + 2.52iT - 11T^{2} \) |
| 13 | \( 1 + 4.10iT - 13T^{2} \) |
| 17 | \( 1 + 4.37T + 17T^{2} \) |
| 19 | \( 1 - 3.46iT - 19T^{2} \) |
| 23 | \( 1 + 8.51iT - 23T^{2} \) |
| 29 | \( 1 - 0.939iT - 29T^{2} \) |
| 31 | \( 1 + 3.46iT - 31T^{2} \) |
| 37 | \( 1 - 6.74T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 4.74T + 43T^{2} \) |
| 47 | \( 1 - 1.62T + 47T^{2} \) |
| 53 | \( 1 - 1.87iT - 53T^{2} \) |
| 59 | \( 1 + 8.74T + 59T^{2} \) |
| 61 | \( 1 + 6.92iT - 61T^{2} \) |
| 67 | \( 1 + 4.74T + 67T^{2} \) |
| 71 | \( 1 - 0.294iT - 71T^{2} \) |
| 73 | \( 1 - 6.92iT - 73T^{2} \) |
| 79 | \( 1 + 2.37T + 79T^{2} \) |
| 83 | \( 1 - 17.4T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 + 11.0iT - 97T^{2} \) |
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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
−10.89048467268808551671247289692, −10.38610923828762090657650993622, −9.028939748744814991564033167835, −8.243978421817344486471213112731, −7.86285366798600350287657257982, −6.41433273282958980863982395942, −5.54141190815059891047079417856, −4.57903511795801412383980605138, −3.08884870290844587796562812934, −2.15558278360804170552666146541,
1.45756697331492100396381759157, 2.23246566136237335555898934150, 3.58518614421039986683783069661, 4.66036025455502599878801756939, 6.41934550863339423358096064731, 7.12089671327371578529796418527, 7.71404109989980366446907831645, 8.967002855571990760509489487900, 9.713086676263963082757410747299, 10.84410133330009901527984626773
