L(s) = 1 | − 1.96·2-s − 2.44·3-s + 1.86·4-s + 4.81·6-s − 2.28·7-s + 0.256·8-s + 2.98·9-s + 4.45·11-s − 4.57·12-s + 2.62·13-s + 4.48·14-s − 4.24·16-s + 5.62·17-s − 5.87·18-s − 4.92·19-s + 5.58·21-s − 8.76·22-s + 2.14·23-s − 0.626·24-s − 5.17·26-s + 0.0375·27-s − 4.26·28-s − 1.59·29-s − 3.29·31-s + 7.83·32-s − 10.9·33-s − 11.0·34-s + ⋯ |
L(s) = 1 | − 1.39·2-s − 1.41·3-s + 0.934·4-s + 1.96·6-s − 0.862·7-s + 0.0905·8-s + 0.994·9-s + 1.34·11-s − 1.32·12-s + 0.729·13-s + 1.19·14-s − 1.06·16-s + 1.36·17-s − 1.38·18-s − 1.13·19-s + 1.21·21-s − 1.86·22-s + 0.446·23-s − 0.127·24-s − 1.01·26-s + 0.00722·27-s − 0.806·28-s − 0.297·29-s − 0.591·31-s + 1.38·32-s − 1.89·33-s − 1.89·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 241 | \( 1 + T \) |
good | 2 | \( 1 + 1.96T + 2T^{2} \) |
| 3 | \( 1 + 2.44T + 3T^{2} \) |
| 7 | \( 1 + 2.28T + 7T^{2} \) |
| 11 | \( 1 - 4.45T + 11T^{2} \) |
| 13 | \( 1 - 2.62T + 13T^{2} \) |
| 17 | \( 1 - 5.62T + 17T^{2} \) |
| 19 | \( 1 + 4.92T + 19T^{2} \) |
| 23 | \( 1 - 2.14T + 23T^{2} \) |
| 29 | \( 1 + 1.59T + 29T^{2} \) |
| 31 | \( 1 + 3.29T + 31T^{2} \) |
| 37 | \( 1 - 3.32T + 37T^{2} \) |
| 41 | \( 1 - 4.94T + 41T^{2} \) |
| 43 | \( 1 + 5.48T + 43T^{2} \) |
| 47 | \( 1 + 9.60T + 47T^{2} \) |
| 53 | \( 1 + 4.43T + 53T^{2} \) |
| 59 | \( 1 + 2.35T + 59T^{2} \) |
| 61 | \( 1 - 7.22T + 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 - 3.03T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 + 7.36T + 79T^{2} \) |
| 83 | \( 1 + 1.30T + 83T^{2} \) |
| 89 | \( 1 - 14.3T + 89T^{2} \) |
| 97 | \( 1 + 8.92T + 97T^{2} \) |
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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
−7.71507867841316258558613758989, −6.96248247086946119971313828254, −6.31675506069900507183606254427, −6.02889378383302408775113531075, −4.95542361524039038950340801036, −4.08972215812174837885884178319, −3.20466038823142273359653699268, −1.67944639814537746434409656694, −0.975704156680067257571672378654, 0,
0.975704156680067257571672378654, 1.67944639814537746434409656694, 3.20466038823142273359653699268, 4.08972215812174837885884178319, 4.95542361524039038950340801036, 6.02889378383302408775113531075, 6.31675506069900507183606254427, 6.96248247086946119971313828254, 7.71507867841316258558613758989