| L(s) = 1 | − 2.48·2-s + 1.61·3-s + 4.15·4-s − 3.04·5-s − 4.01·6-s + 0.428·7-s − 5.35·8-s − 0.381·9-s + 7.54·10-s + 4.36·11-s + 6.73·12-s − 3.80·13-s − 1.06·14-s − 4.92·15-s + 4.97·16-s − 17-s + 0.945·18-s + 3.99·19-s − 12.6·20-s + 0.693·21-s − 10.8·22-s + 4.55·23-s − 8.66·24-s + 4.25·25-s + 9.44·26-s − 5.47·27-s + 1.78·28-s + ⋯ |
| L(s) = 1 | − 1.75·2-s + 0.934·3-s + 2.07·4-s − 1.36·5-s − 1.63·6-s + 0.162·7-s − 1.89·8-s − 0.127·9-s + 2.38·10-s + 1.31·11-s + 1.94·12-s − 1.05·13-s − 0.284·14-s − 1.27·15-s + 1.24·16-s − 0.242·17-s + 0.222·18-s + 0.916·19-s − 2.82·20-s + 0.151·21-s − 2.30·22-s + 0.950·23-s − 1.76·24-s + 0.851·25-s + 1.85·26-s − 1.05·27-s + 0.337·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7911345335\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7911345335\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + T \) |
| 353 | \( 1 - T \) |
| good | 2 | \( 1 + 2.48T + 2T^{2} \) |
| 3 | \( 1 - 1.61T + 3T^{2} \) |
| 5 | \( 1 + 3.04T + 5T^{2} \) |
| 7 | \( 1 - 0.428T + 7T^{2} \) |
| 11 | \( 1 - 4.36T + 11T^{2} \) |
| 13 | \( 1 + 3.80T + 13T^{2} \) |
| 19 | \( 1 - 3.99T + 19T^{2} \) |
| 23 | \( 1 - 4.55T + 23T^{2} \) |
| 29 | \( 1 - 9.82T + 29T^{2} \) |
| 31 | \( 1 - 3.22T + 31T^{2} \) |
| 37 | \( 1 + 3.81T + 37T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 - 1.75T + 43T^{2} \) |
| 47 | \( 1 - 0.449T + 47T^{2} \) |
| 53 | \( 1 + 1.26T + 53T^{2} \) |
| 59 | \( 1 - 13.3T + 59T^{2} \) |
| 61 | \( 1 - 3.54T + 61T^{2} \) |
| 67 | \( 1 - 4.54T + 67T^{2} \) |
| 71 | \( 1 + 8.92T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 + 2.58T + 79T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 + 0.593T + 97T^{2} \) |
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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
−8.093656337338011866835547804300, −7.79449381829543324469533141240, −6.86613594022732760859478221611, −6.71521312165345992591067291105, −5.16029397158493993912834430226, −4.21574437491398446978789059301, −3.26817696843726905729739635158, −2.70606984963463200806924619460, −1.58380112600027682263216024486, −0.59735534228008713751764692558,
0.59735534228008713751764692558, 1.58380112600027682263216024486, 2.70606984963463200806924619460, 3.26817696843726905729739635158, 4.21574437491398446978789059301, 5.16029397158493993912834430226, 6.71521312165345992591067291105, 6.86613594022732760859478221611, 7.79449381829543324469533141240, 8.093656337338011866835547804300
