L(s) = 1 | + 2.41·2-s + 3.82·4-s + 0.585·5-s − 3.41·7-s + 4.41·8-s + 1.41·10-s − 2.82·11-s − 8.24·14-s + 2.99·16-s + 7.41·17-s − 6.24·19-s + 2.24·20-s − 6.82·22-s + 23-s − 4.65·25-s − 13.0·28-s + 8.48·29-s + 8.48·31-s − 1.58·32-s + 17.8·34-s − 2·35-s − 4.82·37-s − 15.0·38-s + 2.58·40-s − 1.65·41-s − 1.75·43-s − 10.8·44-s + ⋯ |
L(s) = 1 | + 1.70·2-s + 1.91·4-s + 0.261·5-s − 1.29·7-s + 1.56·8-s + 0.447·10-s − 0.852·11-s − 2.20·14-s + 0.749·16-s + 1.79·17-s − 1.43·19-s + 0.501·20-s − 1.45·22-s + 0.208·23-s − 0.931·25-s − 2.47·28-s + 1.57·29-s + 1.52·31-s − 0.280·32-s + 3.06·34-s − 0.338·35-s − 0.793·37-s − 2.44·38-s + 0.408·40-s − 0.258·41-s − 0.267·43-s − 1.63·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.645501427\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.645501427\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 - 2.41T + 2T^{2} \) |
| 5 | \( 1 - 0.585T + 5T^{2} \) |
| 7 | \( 1 + 3.41T + 7T^{2} \) |
| 11 | \( 1 + 2.82T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 7.41T + 17T^{2} \) |
| 19 | \( 1 + 6.24T + 19T^{2} \) |
| 29 | \( 1 - 8.48T + 29T^{2} \) |
| 31 | \( 1 - 8.48T + 31T^{2} \) |
| 37 | \( 1 + 4.82T + 37T^{2} \) |
| 41 | \( 1 + 1.65T + 41T^{2} \) |
| 43 | \( 1 + 1.75T + 43T^{2} \) |
| 47 | \( 1 + 0.343T + 47T^{2} \) |
| 53 | \( 1 + 5.07T + 53T^{2} \) |
| 59 | \( 1 - 7.65T + 59T^{2} \) |
| 61 | \( 1 + 0.828T + 61T^{2} \) |
| 67 | \( 1 - 8.58T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 - 13.3T + 73T^{2} \) |
| 79 | \( 1 - 7.89T + 79T^{2} \) |
| 83 | \( 1 - 6.82T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−12.53143303684974527068627997510, −12.00574886498953030043687201754, −10.55121913261116891635483991909, −9.841814834630124358561652039511, −8.140672538903766441421366811653, −6.72003367824527123537023177594, −6.01545013893860363354515947754, −4.96020071882048629169365567112, −3.60321318675161354972819179117, −2.62931122275476910363783739075,
2.62931122275476910363783739075, 3.60321318675161354972819179117, 4.96020071882048629169365567112, 6.01545013893860363354515947754, 6.72003367824527123537023177594, 8.140672538903766441421366811653, 9.841814834630124358561652039511, 10.55121913261116891635483991909, 12.00574886498953030043687201754, 12.53143303684974527068627997510