L(s) = 1 | − 2-s − 3-s − 4-s − 2·5-s + 6-s + 3·8-s + 9-s + 2·10-s + 2.47·11-s + 12-s − 2·13-s + 2·15-s − 16-s − 4.47·17-s − 18-s + 2.47·19-s + 2·20-s − 2.47·22-s − 23-s − 3·24-s − 25-s + 2·26-s − 27-s + 29-s − 2·30-s − 8·31-s − 5·32-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s − 0.5·4-s − 0.894·5-s + 0.408·6-s + 1.06·8-s + 0.333·9-s + 0.632·10-s + 0.745·11-s + 0.288·12-s − 0.554·13-s + 0.516·15-s − 0.250·16-s − 1.08·17-s − 0.235·18-s + 0.567·19-s + 0.447·20-s − 0.527·22-s − 0.208·23-s − 0.612·24-s − 0.200·25-s + 0.392·26-s − 0.192·27-s + 0.185·29-s − 0.365·30-s − 1.43·31-s − 0.883·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4488698067\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4488698067\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
good | 2 | \( 1 + T + 2T^{2} \) |
| 5 | \( 1 + 2T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 2.47T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 4.47T + 17T^{2} \) |
| 19 | \( 1 - 2.47T + 19T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 4.47T + 37T^{2} \) |
| 41 | \( 1 - 6.94T + 41T^{2} \) |
| 43 | \( 1 + 2.47T + 43T^{2} \) |
| 47 | \( 1 + 4.94T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 3.52T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 6.94T + 73T^{2} \) |
| 79 | \( 1 - 1.52T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - 13.4T + 89T^{2} \) |
| 97 | \( 1 + 9.41T + 97T^{2} \) |
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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
−9.338447511055055077126074365178, −8.362387586883570179645531607502, −7.66838139848996064386030734580, −7.04343517956396977846239193971, −6.08054193823677989389679726765, −4.94425588189532455254938780655, −4.34074582951104387990750663525, −3.51216822143666048679181254394, −1.84492219283261523096541628336, −0.51485693571527491062728122169,
0.51485693571527491062728122169, 1.84492219283261523096541628336, 3.51216822143666048679181254394, 4.34074582951104387990750663525, 4.94425588189532455254938780655, 6.08054193823677989389679726765, 7.04343517956396977846239193971, 7.66838139848996064386030734580, 8.362387586883570179645531607502, 9.338447511055055077126074365178