L(s) = 1 | − 2.68·2-s − 3-s + 5.19·4-s − 0.525·5-s + 2.68·6-s + 1.96·7-s − 8.58·8-s + 9-s + 1.40·10-s + 4.22·11-s − 5.19·12-s − 5.20·13-s − 5.28·14-s + 0.525·15-s + 12.6·16-s + 17-s − 2.68·18-s − 2.46·19-s − 2.73·20-s − 1.96·21-s − 11.3·22-s − 3.47·23-s + 8.58·24-s − 4.72·25-s + 13.9·26-s − 27-s + 10.2·28-s + ⋯ |
L(s) = 1 | − 1.89·2-s − 0.577·3-s + 2.59·4-s − 0.234·5-s + 1.09·6-s + 0.744·7-s − 3.03·8-s + 0.333·9-s + 0.445·10-s + 1.27·11-s − 1.50·12-s − 1.44·13-s − 1.41·14-s + 0.135·15-s + 3.15·16-s + 0.242·17-s − 0.632·18-s − 0.565·19-s − 0.610·20-s − 0.429·21-s − 2.41·22-s − 0.725·23-s + 1.75·24-s − 0.944·25-s + 2.73·26-s − 0.192·27-s + 1.93·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 - T \) |
good | 2 | \( 1 + 2.68T + 2T^{2} \) |
| 5 | \( 1 + 0.525T + 5T^{2} \) |
| 7 | \( 1 - 1.96T + 7T^{2} \) |
| 11 | \( 1 - 4.22T + 11T^{2} \) |
| 13 | \( 1 + 5.20T + 13T^{2} \) |
| 19 | \( 1 + 2.46T + 19T^{2} \) |
| 23 | \( 1 + 3.47T + 23T^{2} \) |
| 29 | \( 1 - 6.01T + 29T^{2} \) |
| 31 | \( 1 + 1.06T + 31T^{2} \) |
| 37 | \( 1 + 2.45T + 37T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 + 1.10T + 43T^{2} \) |
| 47 | \( 1 - 3.91T + 47T^{2} \) |
| 53 | \( 1 + 0.313T + 53T^{2} \) |
| 59 | \( 1 - 9.87T + 59T^{2} \) |
| 61 | \( 1 + 1.73T + 61T^{2} \) |
| 67 | \( 1 + 0.944T + 67T^{2} \) |
| 71 | \( 1 + 4.49T + 71T^{2} \) |
| 73 | \( 1 + 5.99T + 73T^{2} \) |
| 79 | \( 1 - 14.0T + 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 + 2.39T + 89T^{2} \) |
| 97 | \( 1 + 3.48T + 97T^{2} \) |
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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
−7.59718675526930414426682470577, −7.07380726749721119528540486013, −6.38549065007413835658923267887, −5.75217123221312351378999694018, −4.70181752998404770793303785497, −3.87272964422758633180750210427, −2.58243813190312615312378068769, −1.86245433470806011502316294610, −1.02942543226846758077244451635, 0,
1.02942543226846758077244451635, 1.86245433470806011502316294610, 2.58243813190312615312378068769, 3.87272964422758633180750210427, 4.70181752998404770793303785497, 5.75217123221312351378999694018, 6.38549065007413835658923267887, 7.07380726749721119528540486013, 7.59718675526930414426682470577