| L(s) = 1 | − 0.149·2-s − 1.36·3-s − 1.97·4-s + 3.17·5-s + 0.203·6-s − 3.47·7-s + 0.594·8-s − 1.14·9-s − 0.474·10-s + 0.0807·11-s + 2.69·12-s + 5.65·13-s + 0.520·14-s − 4.31·15-s + 3.86·16-s − 5.35·17-s + 0.171·18-s + 3.63·19-s − 6.27·20-s + 4.73·21-s − 0.0120·22-s − 1.35·23-s − 0.809·24-s + 5.05·25-s − 0.846·26-s + 5.64·27-s + 6.87·28-s + ⋯ |
| L(s) = 1 | − 0.105·2-s − 0.786·3-s − 0.988·4-s + 1.41·5-s + 0.0831·6-s − 1.31·7-s + 0.210·8-s − 0.381·9-s − 0.149·10-s + 0.0243·11-s + 0.777·12-s + 1.56·13-s + 0.138·14-s − 1.11·15-s + 0.966·16-s − 1.29·17-s + 0.0403·18-s + 0.834·19-s − 1.40·20-s + 1.03·21-s − 0.00257·22-s − 0.281·23-s − 0.165·24-s + 1.01·25-s − 0.165·26-s + 1.08·27-s + 1.30·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 43 | \( 1 \) |
| good | 2 | \( 1 + 0.149T + 2T^{2} \) |
| 3 | \( 1 + 1.36T + 3T^{2} \) |
| 5 | \( 1 - 3.17T + 5T^{2} \) |
| 7 | \( 1 + 3.47T + 7T^{2} \) |
| 11 | \( 1 - 0.0807T + 11T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 17 | \( 1 + 5.35T + 17T^{2} \) |
| 19 | \( 1 - 3.63T + 19T^{2} \) |
| 23 | \( 1 + 1.35T + 23T^{2} \) |
| 29 | \( 1 + 1.06T + 29T^{2} \) |
| 31 | \( 1 + 0.781T + 31T^{2} \) |
| 37 | \( 1 - 1.32T + 37T^{2} \) |
| 41 | \( 1 + 6.42T + 41T^{2} \) |
| 47 | \( 1 + 6.22T + 47T^{2} \) |
| 53 | \( 1 - 10.2T + 53T^{2} \) |
| 59 | \( 1 + 14.4T + 59T^{2} \) |
| 61 | \( 1 - 9.46T + 61T^{2} \) |
| 67 | \( 1 + 2.95T + 67T^{2} \) |
| 71 | \( 1 + 3.42T + 71T^{2} \) |
| 73 | \( 1 + 8.79T + 73T^{2} \) |
| 79 | \( 1 + 6.49T + 79T^{2} \) |
| 83 | \( 1 + 12.0T + 83T^{2} \) |
| 89 | \( 1 - 5.99T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−8.966251261370652096352921358614, −8.457071928483509737707211588345, −6.94321550473988341759830877109, −6.12767333630405877333682250075, −5.85228608514225112882855132834, −4.99282529832264757252529581805, −3.84567469942284225257929988863, −2.87638987800166181727404417818, −1.37137506720001875201777278798, 0,
1.37137506720001875201777278798, 2.87638987800166181727404417818, 3.84567469942284225257929988863, 4.99282529832264757252529581805, 5.85228608514225112882855132834, 6.12767333630405877333682250075, 6.94321550473988341759830877109, 8.457071928483509737707211588345, 8.966251261370652096352921358614
