L(s) = 1 | − 1.59·3-s + 0.804·5-s − 7-s − 0.458·9-s + 11-s + 13-s − 1.28·15-s − 0.398·17-s − 1.98·19-s + 1.59·21-s − 3.65·23-s − 4.35·25-s + 5.51·27-s − 0.285·29-s + 3.90·31-s − 1.59·33-s − 0.804·35-s − 11.0·37-s − 1.59·39-s + 4.88·41-s + 1.43·43-s − 0.368·45-s + 0.275·47-s + 49-s + 0.634·51-s − 3.81·53-s + 0.804·55-s + ⋯ |
L(s) = 1 | − 0.920·3-s + 0.359·5-s − 0.377·7-s − 0.152·9-s + 0.301·11-s + 0.277·13-s − 0.331·15-s − 0.0965·17-s − 0.455·19-s + 0.347·21-s − 0.761·23-s − 0.870·25-s + 1.06·27-s − 0.0530·29-s + 0.701·31-s − 0.277·33-s − 0.135·35-s − 1.81·37-s − 0.255·39-s + 0.762·41-s + 0.219·43-s − 0.0549·45-s + 0.0402·47-s + 0.142·49-s + 0.0888·51-s − 0.523·53-s + 0.108·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9788859308\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9788859308\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 3 | \( 1 + 1.59T + 3T^{2} \) |
| 5 | \( 1 - 0.804T + 5T^{2} \) |
| 17 | \( 1 + 0.398T + 17T^{2} \) |
| 19 | \( 1 + 1.98T + 19T^{2} \) |
| 23 | \( 1 + 3.65T + 23T^{2} \) |
| 29 | \( 1 + 0.285T + 29T^{2} \) |
| 31 | \( 1 - 3.90T + 31T^{2} \) |
| 37 | \( 1 + 11.0T + 37T^{2} \) |
| 41 | \( 1 - 4.88T + 41T^{2} \) |
| 43 | \( 1 - 1.43T + 43T^{2} \) |
| 47 | \( 1 - 0.275T + 47T^{2} \) |
| 53 | \( 1 + 3.81T + 53T^{2} \) |
| 59 | \( 1 - 8.19T + 59T^{2} \) |
| 61 | \( 1 + 7.08T + 61T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 + 2.28T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 + 3.20T + 79T^{2} \) |
| 83 | \( 1 - 4.69T + 83T^{2} \) |
| 89 | \( 1 + 2.84T + 89T^{2} \) |
| 97 | \( 1 - 2.50T + 97T^{2} \) |
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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
−7.88225411719757555191658472876, −6.78386798992178839916544971276, −6.47952494146792575211324782826, −5.72698673278601147098927069274, −5.28567013843178683344509500589, −4.31985673511933413164120291146, −3.62422894824313690197132829375, −2.61098232213773353702685824437, −1.69453497619281752773748076084, −0.50868390065528439472583494855,
0.50868390065528439472583494855, 1.69453497619281752773748076084, 2.61098232213773353702685824437, 3.62422894824313690197132829375, 4.31985673511933413164120291146, 5.28567013843178683344509500589, 5.72698673278601147098927069274, 6.47952494146792575211324782826, 6.78386798992178839916544971276, 7.88225411719757555191658472876