| L(s) = 1 | − 1.30·2-s − 0.296·4-s − 0.572·5-s + 1.21·7-s + 2.99·8-s + 0.747·10-s + 3.82·11-s + 1.55·13-s − 1.58·14-s − 3.31·16-s − 3.24·17-s + 7.65·19-s + 0.169·20-s − 4.98·22-s + 23-s − 4.67·25-s − 2.03·26-s − 0.361·28-s − 29-s − 9.54·31-s − 1.66·32-s + 4.23·34-s − 0.697·35-s − 5.96·37-s − 9.99·38-s − 1.71·40-s + 2.82·41-s + ⋯ |
| L(s) = 1 | − 0.922·2-s − 0.148·4-s − 0.256·5-s + 0.460·7-s + 1.05·8-s + 0.236·10-s + 1.15·11-s + 0.431·13-s − 0.424·14-s − 0.829·16-s − 0.786·17-s + 1.75·19-s + 0.0380·20-s − 1.06·22-s + 0.208·23-s − 0.934·25-s − 0.398·26-s − 0.0682·28-s − 0.185·29-s − 1.71·31-s − 0.294·32-s + 0.726·34-s − 0.117·35-s − 0.981·37-s − 1.62·38-s − 0.271·40-s + 0.441·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 1.30T + 2T^{2} \) |
| 5 | \( 1 + 0.572T + 5T^{2} \) |
| 7 | \( 1 - 1.21T + 7T^{2} \) |
| 11 | \( 1 - 3.82T + 11T^{2} \) |
| 13 | \( 1 - 1.55T + 13T^{2} \) |
| 17 | \( 1 + 3.24T + 17T^{2} \) |
| 19 | \( 1 - 7.65T + 19T^{2} \) |
| 31 | \( 1 + 9.54T + 31T^{2} \) |
| 37 | \( 1 + 5.96T + 37T^{2} \) |
| 41 | \( 1 - 2.82T + 41T^{2} \) |
| 43 | \( 1 + 11.8T + 43T^{2} \) |
| 47 | \( 1 + 0.392T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 - 13.0T + 59T^{2} \) |
| 61 | \( 1 - 9.25T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 + 2.55T + 73T^{2} \) |
| 79 | \( 1 - 0.852T + 79T^{2} \) |
| 83 | \( 1 - 0.606T + 83T^{2} \) |
| 89 | \( 1 + 3.82T + 89T^{2} \) |
| 97 | \( 1 - 5.13T + 97T^{2} \) |
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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
−7.77703269233316513893408177784, −7.26111433634682126065662136996, −6.55766362819119187694166558569, −5.50840621471266191520102614023, −4.85450597964289543800702172570, −3.95278107300306613340772188467, −3.38759359340805865456448219984, −1.84375243585558926287191079852, −1.28324178134255332085327300845, 0,
1.28324178134255332085327300845, 1.84375243585558926287191079852, 3.38759359340805865456448219984, 3.95278107300306613340772188467, 4.85450597964289543800702172570, 5.50840621471266191520102614023, 6.55766362819119187694166558569, 7.26111433634682126065662136996, 7.77703269233316513893408177784
