L(s) = 1 | − 2.09·2-s − 1.81·3-s + 2.37·4-s + 1.01·5-s + 3.79·6-s + 2.98·7-s − 0.779·8-s + 0.297·9-s − 2.12·10-s − 2.86·11-s − 4.30·12-s + 0.868·13-s − 6.24·14-s − 1.84·15-s − 3.11·16-s + 17-s − 0.622·18-s + 1.72·19-s + 2.40·20-s − 5.41·21-s + 5.99·22-s − 6.20·23-s + 1.41·24-s − 3.97·25-s − 1.81·26-s + 4.90·27-s + 7.08·28-s + ⋯ |
L(s) = 1 | − 1.47·2-s − 1.04·3-s + 1.18·4-s + 0.453·5-s + 1.55·6-s + 1.12·7-s − 0.275·8-s + 0.0992·9-s − 0.671·10-s − 0.864·11-s − 1.24·12-s + 0.240·13-s − 1.66·14-s − 0.475·15-s − 0.778·16-s + 0.242·17-s − 0.146·18-s + 0.394·19-s + 0.538·20-s − 1.18·21-s + 1.27·22-s − 1.29·23-s + 0.288·24-s − 0.794·25-s − 0.356·26-s + 0.944·27-s + 1.33·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 - T \) |
| 353 | \( 1 - T \) |
good | 2 | \( 1 + 2.09T + 2T^{2} \) |
| 3 | \( 1 + 1.81T + 3T^{2} \) |
| 5 | \( 1 - 1.01T + 5T^{2} \) |
| 7 | \( 1 - 2.98T + 7T^{2} \) |
| 11 | \( 1 + 2.86T + 11T^{2} \) |
| 13 | \( 1 - 0.868T + 13T^{2} \) |
| 19 | \( 1 - 1.72T + 19T^{2} \) |
| 23 | \( 1 + 6.20T + 23T^{2} \) |
| 29 | \( 1 + 0.219T + 29T^{2} \) |
| 31 | \( 1 - 5.98T + 31T^{2} \) |
| 37 | \( 1 - 4.78T + 37T^{2} \) |
| 41 | \( 1 - 9.96T + 41T^{2} \) |
| 43 | \( 1 - 3.15T + 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 - 4.10T + 53T^{2} \) |
| 59 | \( 1 + 12.2T + 59T^{2} \) |
| 61 | \( 1 - 3.65T + 61T^{2} \) |
| 67 | \( 1 + 11.9T + 67T^{2} \) |
| 71 | \( 1 + 5.83T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 - 9.40T + 79T^{2} \) |
| 83 | \( 1 + 12.4T + 83T^{2} \) |
| 89 | \( 1 - 3.38T + 89T^{2} \) |
| 97 | \( 1 - 5.45T + 97T^{2} \) |
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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
−7.84169389159821476403341898540, −7.36553352091915683738166390525, −6.13661150553922892006365681298, −5.92220084487358997858609572469, −4.92786584217755813048871100368, −4.36349737292495317321092276500, −2.79250483580217813704886378112, −1.88566283800054591839746116043, −1.06676812744208658062857881651, 0,
1.06676812744208658062857881651, 1.88566283800054591839746116043, 2.79250483580217813704886378112, 4.36349737292495317321092276500, 4.92786584217755813048871100368, 5.92220084487358997858609572469, 6.13661150553922892006365681298, 7.36553352091915683738166390525, 7.84169389159821476403341898540