L(s) = 1 | + 2-s − 3-s + 4-s − 2·5-s − 6-s + 8-s + 9-s − 2·10-s + 4.89·11-s − 12-s + 6.89·13-s + 2·15-s + 16-s + 17-s + 18-s + 4.89·19-s − 2·20-s + 4.89·22-s − 4·23-s − 24-s − 25-s + 6.89·26-s − 27-s − 2.89·29-s + 2·30-s − 9.79·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.894·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.632·10-s + 1.47·11-s − 0.288·12-s + 1.91·13-s + 0.516·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s + 1.12·19-s − 0.447·20-s + 1.04·22-s − 0.834·23-s − 0.204·24-s − 0.200·25-s + 1.35·26-s − 0.192·27-s − 0.538·29-s + 0.365·30-s − 1.75·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4998 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4998 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.679857329\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.679857329\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 + 2T + 5T^{2} \) |
| 11 | \( 1 - 4.89T + 11T^{2} \) |
| 13 | \( 1 - 6.89T + 13T^{2} \) |
| 19 | \( 1 - 4.89T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 2.89T + 29T^{2} \) |
| 31 | \( 1 + 9.79T + 31T^{2} \) |
| 37 | \( 1 - 2.89T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 8.89T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 13.7T + 67T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + 9.79T + 79T^{2} \) |
| 83 | \( 1 - 16.8T + 83T^{2} \) |
| 89 | \( 1 + 7.79T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−8.048160341246860124827339522444, −7.38835308800461822975174726593, −6.71224424019762898437793836126, −5.86030450032753941446766265824, −5.58233602789812176213175296577, −4.19683615823150487019048295737, −3.93364664004484454386709160798, −3.32114428254338787724449011062, −1.75766201129522636553993355672, −0.890761286572493024589272891974,
0.890761286572493024589272891974, 1.75766201129522636553993355672, 3.32114428254338787724449011062, 3.93364664004484454386709160798, 4.19683615823150487019048295737, 5.58233602789812176213175296577, 5.86030450032753941446766265824, 6.71224424019762898437793836126, 7.38835308800461822975174726593, 8.048160341246860124827339522444