L(s) = 1 | + 0.0666·3-s + 3.14·5-s − 2.33·7-s − 2.99·9-s + 5.64·11-s + 13-s + 0.209·15-s − 4.41·17-s + 1.84·19-s − 0.155·21-s + 4.97·23-s + 4.90·25-s − 0.399·27-s − 29-s + 5.92·31-s + 0.376·33-s − 7.35·35-s + 6.22·37-s + 0.0666·39-s − 10.9·41-s + 5.61·43-s − 9.42·45-s + 8.89·47-s − 1.53·49-s − 0.294·51-s − 13.8·53-s + 17.7·55-s + ⋯ |
L(s) = 1 | + 0.0384·3-s + 1.40·5-s − 0.883·7-s − 0.998·9-s + 1.70·11-s + 0.277·13-s + 0.0541·15-s − 1.07·17-s + 0.422·19-s − 0.0340·21-s + 1.03·23-s + 0.980·25-s − 0.0769·27-s − 0.185·29-s + 1.06·31-s + 0.0655·33-s − 1.24·35-s + 1.02·37-s + 0.0106·39-s − 1.71·41-s + 0.855·43-s − 1.40·45-s + 1.29·47-s − 0.219·49-s − 0.0412·51-s − 1.89·53-s + 2.39·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.533227997\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.533227997\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 13 | \( 1 - T \) |
| 29 | \( 1 + T \) |
good | 3 | \( 1 - 0.0666T + 3T^{2} \) |
| 5 | \( 1 - 3.14T + 5T^{2} \) |
| 7 | \( 1 + 2.33T + 7T^{2} \) |
| 11 | \( 1 - 5.64T + 11T^{2} \) |
| 17 | \( 1 + 4.41T + 17T^{2} \) |
| 19 | \( 1 - 1.84T + 19T^{2} \) |
| 23 | \( 1 - 4.97T + 23T^{2} \) |
| 31 | \( 1 - 5.92T + 31T^{2} \) |
| 37 | \( 1 - 6.22T + 37T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 - 5.61T + 43T^{2} \) |
| 47 | \( 1 - 8.89T + 47T^{2} \) |
| 53 | \( 1 + 13.8T + 53T^{2} \) |
| 59 | \( 1 + 2.54T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 + 0.815T + 67T^{2} \) |
| 71 | \( 1 - 0.792T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 - 0.342T + 79T^{2} \) |
| 83 | \( 1 + 8.95T + 83T^{2} \) |
| 89 | \( 1 + 0.194T + 89T^{2} \) |
| 97 | \( 1 - 14.5T + 97T^{2} \) |
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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
−8.326643268605796202037295310940, −7.06519840780837790798169343112, −6.39423866863459745087899980044, −6.21490823070733204278336004370, −5.37359348501588452539383724617, −4.47811750127632626863714147751, −3.47371019712010140183656977784, −2.77857792973050967496328623084, −1.89128175344365772275890696884, −0.850506737254918407583931777681,
0.850506737254918407583931777681, 1.89128175344365772275890696884, 2.77857792973050967496328623084, 3.47371019712010140183656977784, 4.47811750127632626863714147751, 5.37359348501588452539383724617, 6.21490823070733204278336004370, 6.39423866863459745087899980044, 7.06519840780837790798169343112, 8.326643268605796202037295310940