L(s) = 1 | + 2.64·2-s + 2.89·3-s + 5.00·4-s − 0.0576·5-s + 7.67·6-s − 4.69·7-s + 7.94·8-s + 5.40·9-s − 0.152·10-s − 2.89·11-s + 14.5·12-s + 1.05·13-s − 12.4·14-s − 0.167·15-s + 11.0·16-s − 17-s + 14.3·18-s − 6.13·19-s − 0.288·20-s − 13.6·21-s − 7.67·22-s − 1.09·23-s + 23.0·24-s − 4.99·25-s + 2.79·26-s + 6.98·27-s − 23.4·28-s + ⋯ |
L(s) = 1 | + 1.87·2-s + 1.67·3-s + 2.50·4-s − 0.0257·5-s + 3.13·6-s − 1.77·7-s + 2.80·8-s + 1.80·9-s − 0.0482·10-s − 0.874·11-s + 4.18·12-s + 0.292·13-s − 3.32·14-s − 0.0431·15-s + 2.75·16-s − 0.242·17-s + 3.37·18-s − 1.40·19-s − 0.0644·20-s − 2.97·21-s − 1.63·22-s − 0.227·23-s + 4.70·24-s − 0.999·25-s + 0.547·26-s + 1.34·27-s − 4.44·28-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
\(L(1)\) |
\(\approx\) |
\(6.784988579\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.784988579\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.64T + 2T^{2} \) |
| 3 | \( 1 - 2.89T + 3T^{2} \) |
| 5 | \( 1 + 0.0576T + 5T^{2} \) |
| 7 | \( 1 + 4.69T + 7T^{2} \) |
| 11 | \( 1 + 2.89T + 11T^{2} \) |
| 13 | \( 1 - 1.05T + 13T^{2} \) |
| 19 | \( 1 + 6.13T + 19T^{2} \) |
| 23 | \( 1 + 1.09T + 23T^{2} \) |
| 29 | \( 1 + 3.98T + 29T^{2} \) |
| 31 | \( 1 - 6.59T + 31T^{2} \) |
| 37 | \( 1 - 3.90T + 37T^{2} \) |
| 41 | \( 1 - 0.303T + 41T^{2} \) |
| 43 | \( 1 - 5.89T + 43T^{2} \) |
| 47 | \( 1 - 9.89T + 47T^{2} \) |
| 53 | \( 1 - 14.1T + 53T^{2} \) |
| 61 | \( 1 + 3.42T + 61T^{2} \) |
| 67 | \( 1 - 5.37T + 67T^{2} \) |
| 71 | \( 1 - 3.29T + 71T^{2} \) |
| 73 | \( 1 - 7.44T + 73T^{2} \) |
| 79 | \( 1 + 7.79T + 79T^{2} \) |
| 83 | \( 1 + 9.61T + 83T^{2} \) |
| 89 | \( 1 - 16.8T + 89T^{2} \) |
| 97 | \( 1 + 4.46T + 97T^{2} \) |
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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
−10.06744087119611553858677753843, −9.155738383445901891247788690691, −8.099029425520928322393418407720, −7.23899683334419719504634478517, −6.44186052612664536284234866725, −5.66945571554876511789975186119, −4.13441317476856254798452009440, −3.80455858295328692776650408158, −2.70143569036161562639270824154, −2.36322602006988945780959734512,
2.36322602006988945780959734512, 2.70143569036161562639270824154, 3.80455858295328692776650408158, 4.13441317476856254798452009440, 5.66945571554876511789975186119, 6.44186052612664536284234866725, 7.23899683334419719504634478517, 8.099029425520928322393418407720, 9.155738383445901891247788690691, 10.06744087119611553858677753843