| L(s) = 1 | − 0.524·2-s − 2.05·3-s − 1.72·4-s − 5-s + 1.07·6-s − 2.06·7-s + 1.95·8-s + 1.21·9-s + 0.524·10-s − 4.52·11-s + 3.54·12-s + 3.32·13-s + 1.08·14-s + 2.05·15-s + 2.42·16-s + 0.669·17-s − 0.637·18-s + 6.33·19-s + 1.72·20-s + 4.23·21-s + 2.37·22-s − 6.89·23-s − 4.01·24-s + 25-s − 1.74·26-s + 3.66·27-s + 3.55·28-s + ⋯ |
| L(s) = 1 | − 0.370·2-s − 1.18·3-s − 0.862·4-s − 0.447·5-s + 0.439·6-s − 0.779·7-s + 0.690·8-s + 0.405·9-s + 0.165·10-s − 1.36·11-s + 1.02·12-s + 0.923·13-s + 0.289·14-s + 0.530·15-s + 0.606·16-s + 0.162·17-s − 0.150·18-s + 1.45·19-s + 0.385·20-s + 0.923·21-s + 0.506·22-s − 1.43·23-s − 0.818·24-s + 0.200·25-s − 0.342·26-s + 0.705·27-s + 0.672·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 + 0.524T + 2T^{2} \) |
| 3 | \( 1 + 2.05T + 3T^{2} \) |
| 7 | \( 1 + 2.06T + 7T^{2} \) |
| 11 | \( 1 + 4.52T + 11T^{2} \) |
| 13 | \( 1 - 3.32T + 13T^{2} \) |
| 17 | \( 1 - 0.669T + 17T^{2} \) |
| 19 | \( 1 - 6.33T + 19T^{2} \) |
| 23 | \( 1 + 6.89T + 23T^{2} \) |
| 29 | \( 1 - 1.76T + 29T^{2} \) |
| 31 | \( 1 + 0.343T + 31T^{2} \) |
| 37 | \( 1 - 8.70T + 37T^{2} \) |
| 41 | \( 1 - 3.38T + 41T^{2} \) |
| 43 | \( 1 - 7.14T + 43T^{2} \) |
| 47 | \( 1 + 0.777T + 47T^{2} \) |
| 53 | \( 1 - 0.398T + 53T^{2} \) |
| 59 | \( 1 + 7.88T + 59T^{2} \) |
| 61 | \( 1 - 6.54T + 61T^{2} \) |
| 67 | \( 1 - 3.81T + 67T^{2} \) |
| 71 | \( 1 - 8.97T + 71T^{2} \) |
| 73 | \( 1 + 5.05T + 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 + 13.4T + 83T^{2} \) |
| 89 | \( 1 + 14.2T + 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
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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
−8.738579908432124006560065076701, −7.965354197042291730966616518064, −7.36381081935896508914477331961, −6.11071445875398981734964389642, −5.66686091675147667966911452209, −4.82249525043254348778607027572, −3.92287417298057440980128359954, −2.89640652412510531039682358711, −0.973185990879034011111454527842, 0,
0.973185990879034011111454527842, 2.89640652412510531039682358711, 3.92287417298057440980128359954, 4.82249525043254348778607027572, 5.66686091675147667966911452209, 6.11071445875398981734964389642, 7.36381081935896508914477331961, 7.965354197042291730966616518064, 8.738579908432124006560065076701
