| L(s) = 1 | + (−1.14 − 1.29i)3-s + (−0.933 + 2.03i)5-s + (−0.372 + 2.97i)9-s − 2.01i·11-s + 3.28·13-s + (3.70 − 1.11i)15-s + 1.71i·17-s + 4.80i·19-s + 1.36·23-s + (−3.25 − 3.79i)25-s + (4.29 − 2.92i)27-s − 2.21i·29-s + 3.56i·31-s + (−2.62 + 2.31i)33-s − 11.2i·37-s + ⋯ |
| L(s) = 1 | + (−0.661 − 0.749i)3-s + (−0.417 + 0.908i)5-s + (−0.124 + 0.992i)9-s − 0.608i·11-s + 0.910·13-s + (0.957 − 0.288i)15-s + 0.416i·17-s + 1.10i·19-s + 0.284·23-s + (−0.651 − 0.758i)25-s + (0.826 − 0.563i)27-s − 0.412i·29-s + 0.640i·31-s + (−0.456 + 0.402i)33-s − 1.84i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.135 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.135 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7887608560\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7887608560\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.14 + 1.29i)T \) |
| 5 | \( 1 + (0.933 - 2.03i)T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 2.01iT - 11T^{2} \) |
| 13 | \( 1 - 3.28T + 13T^{2} \) |
| 17 | \( 1 - 1.71iT - 17T^{2} \) |
| 19 | \( 1 - 4.80iT - 19T^{2} \) |
| 23 | \( 1 - 1.36T + 23T^{2} \) |
| 29 | \( 1 + 2.21iT - 29T^{2} \) |
| 31 | \( 1 - 3.56iT - 31T^{2} \) |
| 37 | \( 1 + 11.2iT - 37T^{2} \) |
| 41 | \( 1 + 8.97T + 41T^{2} \) |
| 43 | \( 1 - 4.39iT - 43T^{2} \) |
| 47 | \( 1 + 2.50iT - 47T^{2} \) |
| 53 | \( 1 + 2.08T + 53T^{2} \) |
| 59 | \( 1 - 1.46T + 59T^{2} \) |
| 61 | \( 1 - 12.5iT - 61T^{2} \) |
| 67 | \( 1 + 2.94iT - 67T^{2} \) |
| 71 | \( 1 - 14.8iT - 71T^{2} \) |
| 73 | \( 1 - 1.55T + 73T^{2} \) |
| 79 | \( 1 - 9.62T + 79T^{2} \) |
| 83 | \( 1 - 14.2iT - 83T^{2} \) |
| 89 | \( 1 + 16.2T + 89T^{2} \) |
| 97 | \( 1 + 0.657T + 97T^{2} \) |
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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
−8.623262167660867788378831976561, −8.146671683416091831488633020339, −7.35098425546607619178099831443, −6.66596406268919229062607115709, −6.00403516456499828325211348062, −5.42512232106073432368619295109, −4.10547458997975503834621109049, −3.38120147253863021520476316497, −2.27724704151094752519140957166, −1.15150660244729276030812804464,
0.31389202657960245642013252101, 1.49523196654472255995196635228, 3.10263048828347348143479589486, 3.94999984904759516804296448196, 4.85435984924204634459257231692, 5.10410770108654120534812872264, 6.23917380058229663773319822008, 6.89471195289175377154849303967, 7.911162395644211892211171574438, 8.737841894741075399302626414094
