L(s) = 1 | − 2-s + 3-s + 4-s − 2.01·5-s − 6-s + 0.821·7-s − 8-s + 9-s + 2.01·10-s − 4.01·11-s + 12-s + 13-s − 0.821·14-s − 2.01·15-s + 16-s + 5.69·17-s − 18-s − 7.47·19-s − 2.01·20-s + 0.821·21-s + 4.01·22-s + 6.57·23-s − 24-s − 0.953·25-s − 26-s + 27-s + 0.821·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.899·5-s − 0.408·6-s + 0.310·7-s − 0.353·8-s + 0.333·9-s + 0.636·10-s − 1.21·11-s + 0.288·12-s + 0.277·13-s − 0.219·14-s − 0.519·15-s + 0.250·16-s + 1.38·17-s − 0.235·18-s − 1.71·19-s − 0.449·20-s + 0.179·21-s + 0.855·22-s + 1.37·23-s − 0.204·24-s − 0.190·25-s − 0.196·26-s + 0.192·27-s + 0.155·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.307659903\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.307659903\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
good | 5 | \( 1 + 2.01T + 5T^{2} \) |
| 7 | \( 1 - 0.821T + 7T^{2} \) |
| 11 | \( 1 + 4.01T + 11T^{2} \) |
| 17 | \( 1 - 5.69T + 17T^{2} \) |
| 19 | \( 1 + 7.47T + 19T^{2} \) |
| 23 | \( 1 - 6.57T + 23T^{2} \) |
| 29 | \( 1 - 6.26T + 29T^{2} \) |
| 31 | \( 1 - 3.73T + 31T^{2} \) |
| 37 | \( 1 - 5.38T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 + 6.06T + 43T^{2} \) |
| 47 | \( 1 + 6.22T + 47T^{2} \) |
| 53 | \( 1 + 7.76T + 53T^{2} \) |
| 59 | \( 1 - 1.22T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 - 13.8T + 67T^{2} \) |
| 71 | \( 1 + 12.0T + 71T^{2} \) |
| 73 | \( 1 + 1.91T + 73T^{2} \) |
| 79 | \( 1 + 4.90T + 79T^{2} \) |
| 83 | \( 1 + 8.34T + 83T^{2} \) |
| 89 | \( 1 - 6.85T + 89T^{2} \) |
| 97 | \( 1 + 6.26T + 97T^{2} \) |
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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
−7.995304107020257348465861862881, −7.50178468675149817278418097657, −6.64244606068293807937110309377, −5.90662319759885811382138194124, −4.85846195906904269553510576296, −4.31229805561846301945955352280, −3.19537673141382738652580387580, −2.79185667844712697336865118393, −1.68527863001257833228000380875, −0.61483651948257900722242062610,
0.61483651948257900722242062610, 1.68527863001257833228000380875, 2.79185667844712697336865118393, 3.19537673141382738652580387580, 4.31229805561846301945955352280, 4.85846195906904269553510576296, 5.90662319759885811382138194124, 6.64244606068293807937110309377, 7.50178468675149817278418097657, 7.995304107020257348465861862881