L(s) = 1 | + 2-s + 1.75·3-s + 4-s + 5-s + 1.75·6-s − 0.115·7-s + 8-s + 0.0923·9-s + 10-s − 1.05·11-s + 1.75·12-s − 5.92·13-s − 0.115·14-s + 1.75·15-s + 16-s − 7.65·17-s + 0.0923·18-s − 6.01·19-s + 20-s − 0.203·21-s − 1.05·22-s − 0.155·23-s + 1.75·24-s + 25-s − 5.92·26-s − 5.11·27-s − 0.115·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.01·3-s + 0.5·4-s + 0.447·5-s + 0.717·6-s − 0.0436·7-s + 0.353·8-s + 0.0307·9-s + 0.316·10-s − 0.319·11-s + 0.507·12-s − 1.64·13-s − 0.0308·14-s + 0.454·15-s + 0.250·16-s − 1.85·17-s + 0.0217·18-s − 1.37·19-s + 0.223·20-s − 0.0443·21-s − 0.225·22-s − 0.0325·23-s + 0.358·24-s + 0.200·25-s − 1.16·26-s − 0.984·27-s − 0.0218·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 601 | \( 1 + T \) |
good | 3 | \( 1 - 1.75T + 3T^{2} \) |
| 7 | \( 1 + 0.115T + 7T^{2} \) |
| 11 | \( 1 + 1.05T + 11T^{2} \) |
| 13 | \( 1 + 5.92T + 13T^{2} \) |
| 17 | \( 1 + 7.65T + 17T^{2} \) |
| 19 | \( 1 + 6.01T + 19T^{2} \) |
| 23 | \( 1 + 0.155T + 23T^{2} \) |
| 29 | \( 1 - 3.22T + 29T^{2} \) |
| 31 | \( 1 + 3.14T + 31T^{2} \) |
| 37 | \( 1 - 6.44T + 37T^{2} \) |
| 41 | \( 1 + 0.499T + 41T^{2} \) |
| 43 | \( 1 + 3.20T + 43T^{2} \) |
| 47 | \( 1 - 5.42T + 47T^{2} \) |
| 53 | \( 1 - 4.44T + 53T^{2} \) |
| 59 | \( 1 - 2.13T + 59T^{2} \) |
| 61 | \( 1 - 5.01T + 61T^{2} \) |
| 67 | \( 1 + 15.5T + 67T^{2} \) |
| 71 | \( 1 - 8.80T + 71T^{2} \) |
| 73 | \( 1 + 11.1T + 73T^{2} \) |
| 79 | \( 1 + 2.78T + 79T^{2} \) |
| 83 | \( 1 - 5.34T + 83T^{2} \) |
| 89 | \( 1 + 3.18T + 89T^{2} \) |
| 97 | \( 1 + 7.38T + 97T^{2} \) |
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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
−7.68765571410023164258158444395, −6.96907413087378252342665789354, −6.35463046959182833116124147182, −5.50316559213606439791850789078, −4.62110661450077313064828551564, −4.18980089900586417053728365336, −3.02047656391385184105814670021, −2.38948869987546424625466755598, −2.00171943079084645692501644078, 0,
2.00171943079084645692501644078, 2.38948869987546424625466755598, 3.02047656391385184105814670021, 4.18980089900586417053728365336, 4.62110661450077313064828551564, 5.50316559213606439791850789078, 6.35463046959182833116124147182, 6.96907413087378252342665789354, 7.68765571410023164258158444395