L(s) = 1 | − 2-s + 3-s + 4-s + 2.56·5-s − 6-s − 1.96·7-s − 8-s + 9-s − 2.56·10-s − 1.94·11-s + 12-s + 0.935·13-s + 1.96·14-s + 2.56·15-s + 16-s + 2.08·17-s − 18-s + 2.20·19-s + 2.56·20-s − 1.96·21-s + 1.94·22-s + 23-s − 24-s + 1.56·25-s − 0.935·26-s + 27-s − 1.96·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.14·5-s − 0.408·6-s − 0.742·7-s − 0.353·8-s + 0.333·9-s − 0.810·10-s − 0.587·11-s + 0.288·12-s + 0.259·13-s + 0.524·14-s + 0.661·15-s + 0.250·16-s + 0.506·17-s − 0.235·18-s + 0.505·19-s + 0.572·20-s − 0.428·21-s + 0.415·22-s + 0.208·23-s − 0.204·24-s + 0.312·25-s − 0.183·26-s + 0.192·27-s − 0.371·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.006724117\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.006724117\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
good | 5 | \( 1 - 2.56T + 5T^{2} \) |
| 7 | \( 1 + 1.96T + 7T^{2} \) |
| 11 | \( 1 + 1.94T + 11T^{2} \) |
| 13 | \( 1 - 0.935T + 13T^{2} \) |
| 17 | \( 1 - 2.08T + 17T^{2} \) |
| 19 | \( 1 - 2.20T + 19T^{2} \) |
| 31 | \( 1 - 7.28T + 31T^{2} \) |
| 37 | \( 1 + 4.14T + 37T^{2} \) |
| 41 | \( 1 - 5.59T + 41T^{2} \) |
| 43 | \( 1 - 4.97T + 43T^{2} \) |
| 47 | \( 1 + 2.25T + 47T^{2} \) |
| 53 | \( 1 - 3.28T + 53T^{2} \) |
| 59 | \( 1 + 1.11T + 59T^{2} \) |
| 61 | \( 1 + 8.24T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 - 4.07T + 71T^{2} \) |
| 73 | \( 1 + 6.10T + 73T^{2} \) |
| 79 | \( 1 - 13.0T + 79T^{2} \) |
| 83 | \( 1 - 3.03T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 - 4.74T + 97T^{2} \) |
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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
−8.521796286097531137942354148170, −7.83904803572171423872426625902, −7.07496698039788113786208943120, −6.27962125974705855370218077788, −5.72977129777559518082629690760, −4.77495317189637408024900681106, −3.48455762967086227990975136526, −2.78366587588864850893281696329, −1.99818618069142297602772310706, −0.890616669427389115564061231080,
0.890616669427389115564061231080, 1.99818618069142297602772310706, 2.78366587588864850893281696329, 3.48455762967086227990975136526, 4.77495317189637408024900681106, 5.72977129777559518082629690760, 6.27962125974705855370218077788, 7.07496698039788113786208943120, 7.83904803572171423872426625902, 8.521796286097531137942354148170