| L(s) = 1 | − 1.25·3-s + 2.99·5-s + 2.30·7-s − 1.42·9-s − 3.59·11-s + 0.850·13-s − 3.75·15-s − 1.36·17-s − 8.09·19-s − 2.88·21-s − 1.28·23-s + 3.95·25-s + 5.55·27-s + 1.47·29-s + 3.64·31-s + 4.50·33-s + 6.88·35-s + 9.97·37-s − 1.06·39-s + 2.80·41-s − 1.28·43-s − 4.27·45-s + 3.56·47-s − 1.70·49-s + 1.71·51-s + 7.64·53-s − 10.7·55-s + ⋯ |
| L(s) = 1 | − 0.723·3-s + 1.33·5-s + 0.869·7-s − 0.475·9-s − 1.08·11-s + 0.235·13-s − 0.968·15-s − 0.331·17-s − 1.85·19-s − 0.629·21-s − 0.268·23-s + 0.790·25-s + 1.06·27-s + 0.274·29-s + 0.654·31-s + 0.784·33-s + 1.16·35-s + 1.64·37-s − 0.170·39-s + 0.438·41-s − 0.196·43-s − 0.636·45-s + 0.520·47-s − 0.243·49-s + 0.240·51-s + 1.05·53-s − 1.44·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.726414504\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.726414504\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 + 1.25T + 3T^{2} \) |
| 5 | \( 1 - 2.99T + 5T^{2} \) |
| 7 | \( 1 - 2.30T + 7T^{2} \) |
| 11 | \( 1 + 3.59T + 11T^{2} \) |
| 13 | \( 1 - 0.850T + 13T^{2} \) |
| 17 | \( 1 + 1.36T + 17T^{2} \) |
| 19 | \( 1 + 8.09T + 19T^{2} \) |
| 23 | \( 1 + 1.28T + 23T^{2} \) |
| 29 | \( 1 - 1.47T + 29T^{2} \) |
| 31 | \( 1 - 3.64T + 31T^{2} \) |
| 37 | \( 1 - 9.97T + 37T^{2} \) |
| 41 | \( 1 - 2.80T + 41T^{2} \) |
| 43 | \( 1 + 1.28T + 43T^{2} \) |
| 47 | \( 1 - 3.56T + 47T^{2} \) |
| 53 | \( 1 - 7.64T + 53T^{2} \) |
| 59 | \( 1 - 1.08T + 59T^{2} \) |
| 61 | \( 1 - 9.14T + 61T^{2} \) |
| 67 | \( 1 - 13.0T + 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 - 16.7T + 73T^{2} \) |
| 79 | \( 1 + 1.49T + 79T^{2} \) |
| 83 | \( 1 - 8.41T + 83T^{2} \) |
| 89 | \( 1 - 1.78T + 89T^{2} \) |
| 97 | \( 1 - 0.0808T + 97T^{2} \) |
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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
−8.345784552344928412032271166473, −7.932084560869069765919630929362, −6.56752947261656090775938383371, −6.26877991867775084816231884072, −5.40520984458835623366233630453, −5.00397330155157832059300758043, −4.06594234923861883968475686637, −2.48133573220758845864610424053, −2.18144495341612218002921772997, −0.76651230051643605513016213979,
0.76651230051643605513016213979, 2.18144495341612218002921772997, 2.48133573220758845864610424053, 4.06594234923861883968475686637, 5.00397330155157832059300758043, 5.40520984458835623366233630453, 6.26877991867775084816231884072, 6.56752947261656090775938383371, 7.932084560869069765919630929362, 8.345784552344928412032271166473
