L(s) = 1 | + (−2.17 + 0.539i)5-s + (2.17 − 2.17i)7-s + 1.70·11-s + (−4.43 + 4.43i)13-s + (1.63 − 1.63i)17-s + (−2.80 + 3.34i)19-s + (−3.24 − 3.24i)23-s + (4.41 − 2.34i)25-s + 3.27·29-s − 7.11i·31-s + (−3.53 + 5.87i)35-s + (0.128 + 0.128i)37-s + 3.83i·41-s + (−5.24 − 5.24i)43-s + (−0.908 + 0.908i)47-s + ⋯ |
L(s) = 1 | + (−0.970 + 0.241i)5-s + (0.820 − 0.820i)7-s + 0.515·11-s + (−1.23 + 1.23i)13-s + (0.395 − 0.395i)17-s + (−0.642 + 0.766i)19-s + (−0.677 − 0.677i)23-s + (0.883 − 0.468i)25-s + 0.608·29-s − 1.27i·31-s + (−0.598 + 0.993i)35-s + (0.0211 + 0.0211i)37-s + 0.598i·41-s + (−0.800 − 0.800i)43-s + (−0.132 + 0.132i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.845 + 0.533i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.845 + 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4453057674\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4453057674\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.17 - 0.539i)T \) |
| 19 | \( 1 + (2.80 - 3.34i)T \) |
good | 7 | \( 1 + (-2.17 + 2.17i)T - 7iT^{2} \) |
| 11 | \( 1 - 1.70T + 11T^{2} \) |
| 13 | \( 1 + (4.43 - 4.43i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1.63 + 1.63i)T - 17iT^{2} \) |
| 23 | \( 1 + (3.24 + 3.24i)T + 23iT^{2} \) |
| 29 | \( 1 - 3.27T + 29T^{2} \) |
| 31 | \( 1 + 7.11iT - 31T^{2} \) |
| 37 | \( 1 + (-0.128 - 0.128i)T + 37iT^{2} \) |
| 41 | \( 1 - 3.83iT - 41T^{2} \) |
| 43 | \( 1 + (5.24 + 5.24i)T + 43iT^{2} \) |
| 47 | \( 1 + (0.908 - 0.908i)T - 47iT^{2} \) |
| 53 | \( 1 + (-5.47 + 5.47i)T - 53iT^{2} \) |
| 59 | \( 1 + 13.9T + 59T^{2} \) |
| 61 | \( 1 - 2.63T + 61T^{2} \) |
| 67 | \( 1 + (-7.71 - 7.71i)T + 67iT^{2} \) |
| 71 | \( 1 + 1.51iT - 71T^{2} \) |
| 73 | \( 1 + (8.70 + 8.70i)T + 73iT^{2} \) |
| 79 | \( 1 + 4.78T + 79T^{2} \) |
| 83 | \( 1 + (6.46 + 6.46i)T + 83iT^{2} \) |
| 89 | \( 1 + 5.34T + 89T^{2} \) |
| 97 | \( 1 + (10.2 + 10.2i)T + 97iT^{2} \) |
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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
−8.121119915930576338918844685420, −7.59228264189417012158494217030, −6.94872809584379862534995611306, −6.25445753528888767283982597518, −4.95515362175223700306808407318, −4.31030618367118358264905920233, −3.88320707780006349116281428819, −2.59986646481807033023809144329, −1.54493489543497636231891719722, −0.13911864996988207275945532215,
1.30532867775978021906462001611, 2.52661543452200702694700959073, 3.37000567131898561615645050493, 4.42790970238661287743527067949, 5.04823252551231319910986407339, 5.71738294661480842068358303362, 6.83204126635048197917498903514, 7.54741773152548558021923914024, 8.263718007530772622082917402503, 8.620917783234451350033644120274