L(s) = 1 | − 2.20·2-s + 3-s + 2.84·4-s + 2.77·5-s − 2.20·6-s − 1.85·8-s + 9-s − 6.11·10-s − 1.68·11-s + 2.84·12-s + 1.26·13-s + 2.77·15-s − 1.59·16-s − 5.32·17-s − 2.20·18-s − 0.141·19-s + 7.89·20-s + 3.69·22-s − 3.48·23-s − 1.85·24-s + 2.70·25-s − 2.78·26-s + 27-s − 2.99·29-s − 6.11·30-s − 5.66·31-s + 7.23·32-s + ⋯ |
L(s) = 1 | − 1.55·2-s + 0.577·3-s + 1.42·4-s + 1.24·5-s − 0.898·6-s − 0.657·8-s + 0.333·9-s − 1.93·10-s − 0.506·11-s + 0.821·12-s + 0.351·13-s + 0.716·15-s − 0.399·16-s − 1.29·17-s − 0.518·18-s − 0.0325·19-s + 1.76·20-s + 0.788·22-s − 0.725·23-s − 0.379·24-s + 0.541·25-s − 0.547·26-s + 0.192·27-s − 0.555·29-s − 1.11·30-s − 1.01·31-s + 1.27·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
good | 2 | \( 1 + 2.20T + 2T^{2} \) |
| 5 | \( 1 - 2.77T + 5T^{2} \) |
| 11 | \( 1 + 1.68T + 11T^{2} \) |
| 13 | \( 1 - 1.26T + 13T^{2} \) |
| 17 | \( 1 + 5.32T + 17T^{2} \) |
| 19 | \( 1 + 0.141T + 19T^{2} \) |
| 23 | \( 1 + 3.48T + 23T^{2} \) |
| 29 | \( 1 + 2.99T + 29T^{2} \) |
| 31 | \( 1 + 5.66T + 31T^{2} \) |
| 37 | \( 1 - 0.881T + 37T^{2} \) |
| 43 | \( 1 - 4.77T + 43T^{2} \) |
| 47 | \( 1 - 6.78T + 47T^{2} \) |
| 53 | \( 1 + 9.07T + 53T^{2} \) |
| 59 | \( 1 + 3.92T + 59T^{2} \) |
| 61 | \( 1 + 9.47T + 61T^{2} \) |
| 67 | \( 1 - 5.95T + 67T^{2} \) |
| 71 | \( 1 + 2.01T + 71T^{2} \) |
| 73 | \( 1 + 1.94T + 73T^{2} \) |
| 79 | \( 1 + 9.24T + 79T^{2} \) |
| 83 | \( 1 + 0.172T + 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 - 5.03T + 97T^{2} \) |
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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
−7.79493754261778896833396345575, −7.36191768720554546382265470474, −6.43053994744484053405184709332, −5.93310167901267228377523779167, −4.89588495741301954610446309811, −3.92629620075795707982191544522, −2.65851797854198838670715461835, −2.08976157615946192173990072566, −1.41743668688711724449810512499, 0,
1.41743668688711724449810512499, 2.08976157615946192173990072566, 2.65851797854198838670715461835, 3.92629620075795707982191544522, 4.89588495741301954610446309811, 5.93310167901267228377523779167, 6.43053994744484053405184709332, 7.36191768720554546382265470474, 7.79493754261778896833396345575