L(s) = 1 | − 1.21·3-s − 5-s + 7-s − 1.53·9-s + 11-s − 1.21·13-s + 1.21·15-s + 4.27·17-s − 7.06·19-s − 1.21·21-s + 2.95·23-s + 25-s + 5.48·27-s − 2·29-s + 6.16·31-s − 1.21·33-s − 35-s − 4.95·37-s + 1.46·39-s + 0.166·41-s − 6.44·43-s + 1.53·45-s + 2.70·47-s + 49-s − 5.18·51-s + 4.11·53-s − 55-s + ⋯ |
L(s) = 1 | − 0.699·3-s − 0.447·5-s + 0.377·7-s − 0.511·9-s + 0.301·11-s − 0.335·13-s + 0.312·15-s + 1.03·17-s − 1.62·19-s − 0.264·21-s + 0.616·23-s + 0.200·25-s + 1.05·27-s − 0.371·29-s + 1.10·31-s − 0.210·33-s − 0.169·35-s − 0.814·37-s + 0.234·39-s + 0.0259·41-s − 0.982·43-s + 0.228·45-s + 0.393·47-s + 0.142·49-s − 0.725·51-s + 0.564·53-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 | \( 1 + 1.21T + 3T^{2} \) |
| 13 | \( 1 + 1.21T + 13T^{2} \) |
| 17 | \( 1 - 4.27T + 17T^{2} \) |
| 19 | \( 1 + 7.06T + 19T^{2} \) |
| 23 | \( 1 - 2.95T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 6.16T + 31T^{2} \) |
| 37 | \( 1 + 4.95T + 37T^{2} \) |
| 41 | \( 1 - 0.166T + 41T^{2} \) |
| 43 | \( 1 + 6.44T + 43T^{2} \) |
| 47 | \( 1 - 2.70T + 47T^{2} \) |
| 53 | \( 1 - 4.11T + 53T^{2} \) |
| 59 | \( 1 - 14.7T + 59T^{2} \) |
| 61 | \( 1 + 14.3T + 61T^{2} \) |
| 67 | \( 1 - 9.60T + 67T^{2} \) |
| 71 | \( 1 + 6.13T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 - 1.48T + 83T^{2} \) |
| 89 | \( 1 - 2.42T + 89T^{2} \) |
| 97 | \( 1 + 7.71T + 97T^{2} \) |
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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
−7.72184729672029655294569433513, −6.88954559492186336378664122714, −6.31268869311300036962581402023, −5.50859115809739797985054492547, −4.90651111433906934923061444325, −4.14715839200490986527462785506, −3.25967460176208348156636914442, −2.32013302890219695624796532015, −1.12524496262892800779036826500, 0,
1.12524496262892800779036826500, 2.32013302890219695624796532015, 3.25967460176208348156636914442, 4.14715839200490986527462785506, 4.90651111433906934923061444325, 5.50859115809739797985054492547, 6.31268869311300036962581402023, 6.88954559492186336378664122714, 7.72184729672029655294569433513