L(s) = 1 | + 2.26·2-s − 1.91·3-s + 3.11·4-s − 0.621·5-s − 4.34·6-s + 3.74·7-s + 2.52·8-s + 0.686·9-s − 1.40·10-s + 0.321·11-s − 5.98·12-s + 3.48·13-s + 8.46·14-s + 1.19·15-s − 0.525·16-s + 17-s + 1.55·18-s + 2.69·19-s − 1.93·20-s − 7.18·21-s + 0.727·22-s + 6.35·23-s − 4.84·24-s − 4.61·25-s + 7.88·26-s + 4.44·27-s + 11.6·28-s + ⋯ |
L(s) = 1 | + 1.59·2-s − 1.10·3-s + 1.55·4-s − 0.278·5-s − 1.77·6-s + 1.41·7-s + 0.891·8-s + 0.228·9-s − 0.444·10-s + 0.0969·11-s − 1.72·12-s + 0.966·13-s + 2.26·14-s + 0.308·15-s − 0.131·16-s + 0.242·17-s + 0.365·18-s + 0.617·19-s − 0.433·20-s − 1.56·21-s + 0.155·22-s + 1.32·23-s − 0.988·24-s − 0.922·25-s + 1.54·26-s + 0.854·27-s + 2.20·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.056023479\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.056023479\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
good | 2 | \( 1 - 2.26T + 2T^{2} \) |
| 3 | \( 1 + 1.91T + 3T^{2} \) |
| 5 | \( 1 + 0.621T + 5T^{2} \) |
| 7 | \( 1 - 3.74T + 7T^{2} \) |
| 11 | \( 1 - 0.321T + 11T^{2} \) |
| 13 | \( 1 - 3.48T + 13T^{2} \) |
| 19 | \( 1 - 2.69T + 19T^{2} \) |
| 23 | \( 1 - 6.35T + 23T^{2} \) |
| 29 | \( 1 - 3.46T + 29T^{2} \) |
| 31 | \( 1 + 0.773T + 31T^{2} \) |
| 37 | \( 1 - 1.43T + 37T^{2} \) |
| 41 | \( 1 - 8.61T + 41T^{2} \) |
| 43 | \( 1 - 4.87T + 43T^{2} \) |
| 47 | \( 1 + 4.13T + 47T^{2} \) |
| 53 | \( 1 + 1.27T + 53T^{2} \) |
| 61 | \( 1 - 0.0869T + 61T^{2} \) |
| 67 | \( 1 + 13.0T + 67T^{2} \) |
| 71 | \( 1 - 8.33T + 71T^{2} \) |
| 73 | \( 1 + 9.87T + 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 - 1.60T + 83T^{2} \) |
| 89 | \( 1 - 4.51T + 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
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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
−10.61428051461409137705683130409, −9.095281539314319875407889664632, −8.046645835190649431804677715142, −7.11424255130776594374235807271, −6.10560767880115430725453368782, −5.52184842903455158503787830605, −4.80105208533528795556240473221, −4.09143910168999395350583242468, −2.88363566985804930934717612097, −1.29202833063694870607148297538,
1.29202833063694870607148297538, 2.88363566985804930934717612097, 4.09143910168999395350583242468, 4.80105208533528795556240473221, 5.52184842903455158503787830605, 6.10560767880115430725453368782, 7.11424255130776594374235807271, 8.046645835190649431804677715142, 9.095281539314319875407889664632, 10.61428051461409137705683130409