L(s) = 1 | + 2-s + 4-s − 3.28·5-s − 3.44·7-s + 8-s − 3.28·10-s − 2.58·11-s + 0.974·13-s − 3.44·14-s + 16-s + 4.12·17-s − 4.55·19-s − 3.28·20-s − 2.58·22-s − 7.39·23-s + 5.81·25-s + 0.974·26-s − 3.44·28-s − 7.26·29-s + 5.97·31-s + 32-s + 4.12·34-s + 11.3·35-s + 7.38·37-s − 4.55·38-s − 3.28·40-s − 1.89·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.47·5-s − 1.30·7-s + 0.353·8-s − 1.03·10-s − 0.780·11-s + 0.270·13-s − 0.921·14-s + 0.250·16-s + 1.00·17-s − 1.04·19-s − 0.735·20-s − 0.552·22-s − 1.54·23-s + 1.16·25-s + 0.191·26-s − 0.651·28-s − 1.34·29-s + 1.07·31-s + 0.176·32-s + 0.707·34-s + 1.91·35-s + 1.21·37-s − 0.738·38-s − 0.519·40-s − 0.296·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.269425813\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.269425813\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 223 | \( 1 - T \) |
good | 5 | \( 1 + 3.28T + 5T^{2} \) |
| 7 | \( 1 + 3.44T + 7T^{2} \) |
| 11 | \( 1 + 2.58T + 11T^{2} \) |
| 13 | \( 1 - 0.974T + 13T^{2} \) |
| 17 | \( 1 - 4.12T + 17T^{2} \) |
| 19 | \( 1 + 4.55T + 19T^{2} \) |
| 23 | \( 1 + 7.39T + 23T^{2} \) |
| 29 | \( 1 + 7.26T + 29T^{2} \) |
| 31 | \( 1 - 5.97T + 31T^{2} \) |
| 37 | \( 1 - 7.38T + 37T^{2} \) |
| 41 | \( 1 + 1.89T + 41T^{2} \) |
| 43 | \( 1 - 12.5T + 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 - 0.0218T + 53T^{2} \) |
| 59 | \( 1 + 4.13T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 + 5.43T + 67T^{2} \) |
| 71 | \( 1 - 6.53T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 - 14.1T + 83T^{2} \) |
| 89 | \( 1 + 15.5T + 89T^{2} \) |
| 97 | \( 1 + 0.720T + 97T^{2} \) |
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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
−8.027678395020463558571385953572, −7.82673761743878858014143014141, −6.95768849474779172957317884497, −6.12637054714509690506640741688, −5.59180732072969420761329788307, −4.33171973834358727414342030106, −3.92896366321896481381001597816, −3.18739385056813043617859494607, −2.34599701656992027923140804050, −0.54841338919487449272011150716,
0.54841338919487449272011150716, 2.34599701656992027923140804050, 3.18739385056813043617859494607, 3.92896366321896481381001597816, 4.33171973834358727414342030106, 5.59180732072969420761329788307, 6.12637054714509690506640741688, 6.95768849474779172957317884497, 7.82673761743878858014143014141, 8.027678395020463558571385953572