L(s) = 1 | + 0.732·3-s + 5-s + 7-s − 2.46·9-s − 11-s − 3.08·13-s + 0.732·15-s − 3.33·17-s + 6.76·19-s + 0.732·21-s − 2.34·23-s + 25-s − 4·27-s + 3.29·29-s − 8.03·31-s − 0.732·33-s + 35-s + 11.0·37-s − 2.25·39-s − 6.37·41-s + 4.16·43-s − 2.46·45-s − 10.7·47-s + 49-s − 2.44·51-s + 6.50·53-s − 55-s + ⋯ |
L(s) = 1 | + 0.422·3-s + 0.447·5-s + 0.377·7-s − 0.821·9-s − 0.301·11-s − 0.854·13-s + 0.189·15-s − 0.808·17-s + 1.55·19-s + 0.159·21-s − 0.489·23-s + 0.200·25-s − 0.769·27-s + 0.612·29-s − 1.44·31-s − 0.127·33-s + 0.169·35-s + 1.81·37-s − 0.361·39-s − 0.996·41-s + 0.634·43-s − 0.367·45-s − 1.57·47-s + 0.142·49-s − 0.341·51-s + 0.893·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 - 0.732T + 3T^{2} \) |
| 13 | \( 1 + 3.08T + 13T^{2} \) |
| 17 | \( 1 + 3.33T + 17T^{2} \) |
| 19 | \( 1 - 6.76T + 19T^{2} \) |
| 23 | \( 1 + 2.34T + 23T^{2} \) |
| 29 | \( 1 - 3.29T + 29T^{2} \) |
| 31 | \( 1 + 8.03T + 31T^{2} \) |
| 37 | \( 1 - 11.0T + 37T^{2} \) |
| 41 | \( 1 + 6.37T + 41T^{2} \) |
| 43 | \( 1 - 4.16T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 - 6.50T + 53T^{2} \) |
| 59 | \( 1 + 5.61T + 59T^{2} \) |
| 61 | \( 1 - 2.54T + 61T^{2} \) |
| 67 | \( 1 - 0.414T + 67T^{2} \) |
| 71 | \( 1 + 5.46T + 71T^{2} \) |
| 73 | \( 1 + 9.49T + 73T^{2} \) |
| 79 | \( 1 + 3.11T + 79T^{2} \) |
| 83 | \( 1 + 11.6T + 83T^{2} \) |
| 89 | \( 1 + 3.46T + 89T^{2} \) |
| 97 | \( 1 - 8.06T + 97T^{2} \) |
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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
−7.72171334439407017704204068167, −7.16452719828978079027121719448, −6.21695812668831074483006993534, −5.48732694497582652202144207333, −4.95358604426764411188664021619, −4.01848059078737384849974677623, −2.97208948793478894511548302472, −2.47145071706019150292703273172, −1.46997786809645778194346710577, 0,
1.46997786809645778194346710577, 2.47145071706019150292703273172, 2.97208948793478894511548302472, 4.01848059078737384849974677623, 4.95358604426764411188664021619, 5.48732694497582652202144207333, 6.21695812668831074483006993534, 7.16452719828978079027121719448, 7.72171334439407017704204068167