| L(s) = 1 | − 2.39i·3-s − 2.83i·5-s + (2.55 + 0.678i)7-s − 2.74·9-s + (−0.0667 + 3.31i)11-s + 13-s − 6.79·15-s + 4.04·17-s + 4.55·19-s + (1.62 − 6.13i)21-s + 5.77·23-s − 3.02·25-s − 0.607i·27-s + 4.69i·29-s + 1.01i·31-s + ⋯ |
| L(s) = 1 | − 1.38i·3-s − 1.26i·5-s + (0.966 + 0.256i)7-s − 0.915·9-s + (−0.0201 + 0.999i)11-s + 0.277·13-s − 1.75·15-s + 0.980·17-s + 1.04·19-s + (0.354 − 1.33i)21-s + 1.20·23-s − 0.605·25-s − 0.116i·27-s + 0.872i·29-s + 0.182i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.275 + 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.275 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.516272727\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.516272727\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-2.55 - 0.678i)T \) |
| 11 | \( 1 + (0.0667 - 3.31i)T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 2.39iT - 3T^{2} \) |
| 5 | \( 1 + 2.83iT - 5T^{2} \) |
| 17 | \( 1 - 4.04T + 17T^{2} \) |
| 19 | \( 1 - 4.55T + 19T^{2} \) |
| 23 | \( 1 - 5.77T + 23T^{2} \) |
| 29 | \( 1 - 4.69iT - 29T^{2} \) |
| 31 | \( 1 - 1.01iT - 31T^{2} \) |
| 37 | \( 1 + 1.21T + 37T^{2} \) |
| 41 | \( 1 - 0.889T + 41T^{2} \) |
| 43 | \( 1 + 1.03iT - 43T^{2} \) |
| 47 | \( 1 + 4.60iT - 47T^{2} \) |
| 53 | \( 1 + 7.85T + 53T^{2} \) |
| 59 | \( 1 - 2.22iT - 59T^{2} \) |
| 61 | \( 1 - 6.50T + 61T^{2} \) |
| 67 | \( 1 - 8.04T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 + 1.88T + 73T^{2} \) |
| 79 | \( 1 + 11.9iT - 79T^{2} \) |
| 83 | \( 1 - 2.58T + 83T^{2} \) |
| 89 | \( 1 + 14.4iT - 89T^{2} \) |
| 97 | \( 1 - 2.52iT - 97T^{2} \) |
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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
−8.149573899004497096841562996618, −7.48558196849429212595258504944, −7.05306803442188602374408043274, −5.99444660627725019037418431056, −5.03462602066145782433949327842, −4.91727976505680456996304123833, −3.53089487335287655199790549837, −2.28649486970377714373685010094, −1.39272672251008151426007563967, −0.977721007514595125288340883118,
1.09407719282625128041746560085, 2.66657881162496020137334850702, 3.37422071604126726402148151911, 3.91642693645045767912253189257, 4.97834702258973991051382439641, 5.47693745042323318422722866033, 6.38463170945061289088935281887, 7.29386550201782791043727045366, 7.959663477475686184301077186598, 8.696970413964006206286068682959
