| L(s) = 1 | − 2-s − 3-s + 4-s + 2.21·5-s + 6-s − 2.01·7-s − 8-s + 9-s − 2.21·10-s − 3.35·11-s − 12-s + 2.68·13-s + 2.01·14-s − 2.21·15-s + 16-s + 17-s − 18-s + 0.700·19-s + 2.21·20-s + 2.01·21-s + 3.35·22-s − 5.44·23-s + 24-s − 0.0997·25-s − 2.68·26-s − 27-s − 2.01·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.989·5-s + 0.408·6-s − 0.760·7-s − 0.353·8-s + 0.333·9-s − 0.700·10-s − 1.01·11-s − 0.288·12-s + 0.744·13-s + 0.537·14-s − 0.571·15-s + 0.250·16-s + 0.242·17-s − 0.235·18-s + 0.160·19-s + 0.494·20-s + 0.438·21-s + 0.714·22-s − 1.13·23-s + 0.204·24-s − 0.0199·25-s − 0.526·26-s − 0.192·27-s − 0.380·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 5 | \( 1 - 2.21T + 5T^{2} \) |
| 7 | \( 1 + 2.01T + 7T^{2} \) |
| 11 | \( 1 + 3.35T + 11T^{2} \) |
| 13 | \( 1 - 2.68T + 13T^{2} \) |
| 19 | \( 1 - 0.700T + 19T^{2} \) |
| 23 | \( 1 + 5.44T + 23T^{2} \) |
| 29 | \( 1 - 7.80T + 29T^{2} \) |
| 31 | \( 1 - 0.788T + 31T^{2} \) |
| 37 | \( 1 + 7.01T + 37T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 + 2.46T + 43T^{2} \) |
| 47 | \( 1 + 6.98T + 47T^{2} \) |
| 53 | \( 1 - 2.38T + 53T^{2} \) |
| 61 | \( 1 - 8.36T + 61T^{2} \) |
| 67 | \( 1 + 0.815T + 67T^{2} \) |
| 71 | \( 1 + 4.99T + 71T^{2} \) |
| 73 | \( 1 - 6.12T + 73T^{2} \) |
| 79 | \( 1 - 2.65T + 79T^{2} \) |
| 83 | \( 1 - 3.32T + 83T^{2} \) |
| 89 | \( 1 - 7.25T + 89T^{2} \) |
| 97 | \( 1 + 1.21T + 97T^{2} \) |
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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
−7.894121603198953685981081031878, −6.81585613130668568154138029685, −6.38113903967373284071030493548, −5.72087854784947373137384636751, −5.16022936723429678968359487842, −3.99110489693844138654571729794, −2.99194905380910420985342845625, −2.19106006073692698076346359195, −1.19353565007293576667196710842, 0,
1.19353565007293576667196710842, 2.19106006073692698076346359195, 2.99194905380910420985342845625, 3.99110489693844138654571729794, 5.16022936723429678968359487842, 5.72087854784947373137384636751, 6.38113903967373284071030493548, 6.81585613130668568154138029685, 7.894121603198953685981081031878
