L(s) = 1 | + 0.782·3-s + 2.74·5-s + 3.13·7-s − 2.38·9-s − 4.20·11-s + 4.79·13-s + 2.14·15-s − 4.78·17-s + 6.84·19-s + 2.45·21-s − 2.52·23-s + 2.51·25-s − 4.21·27-s − 0.824·29-s + 4.65·31-s − 3.29·33-s + 8.59·35-s − 5.55·37-s + 3.75·39-s + 6.84·41-s + 9.50·43-s − 6.54·45-s + 3.54·47-s + 2.82·49-s − 3.74·51-s + 7.03·53-s − 11.5·55-s + ⋯ |
L(s) = 1 | + 0.451·3-s + 1.22·5-s + 1.18·7-s − 0.795·9-s − 1.26·11-s + 1.33·13-s + 0.553·15-s − 1.16·17-s + 1.57·19-s + 0.535·21-s − 0.525·23-s + 0.503·25-s − 0.811·27-s − 0.153·29-s + 0.836·31-s − 0.573·33-s + 1.45·35-s − 0.912·37-s + 0.600·39-s + 1.06·41-s + 1.44·43-s − 0.976·45-s + 0.517·47-s + 0.403·49-s − 0.524·51-s + 0.966·53-s − 1.55·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.461806580\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.461806580\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.782T + 3T^{2} \) |
| 5 | \( 1 - 2.74T + 5T^{2} \) |
| 7 | \( 1 - 3.13T + 7T^{2} \) |
| 11 | \( 1 + 4.20T + 11T^{2} \) |
| 13 | \( 1 - 4.79T + 13T^{2} \) |
| 17 | \( 1 + 4.78T + 17T^{2} \) |
| 19 | \( 1 - 6.84T + 19T^{2} \) |
| 23 | \( 1 + 2.52T + 23T^{2} \) |
| 29 | \( 1 + 0.824T + 29T^{2} \) |
| 31 | \( 1 - 4.65T + 31T^{2} \) |
| 37 | \( 1 + 5.55T + 37T^{2} \) |
| 41 | \( 1 - 6.84T + 41T^{2} \) |
| 43 | \( 1 - 9.50T + 43T^{2} \) |
| 47 | \( 1 - 3.54T + 47T^{2} \) |
| 53 | \( 1 - 7.03T + 53T^{2} \) |
| 59 | \( 1 - 0.838T + 59T^{2} \) |
| 61 | \( 1 + 1.86T + 61T^{2} \) |
| 67 | \( 1 + 3.63T + 67T^{2} \) |
| 71 | \( 1 - 7.42T + 71T^{2} \) |
| 73 | \( 1 - 3.97T + 73T^{2} \) |
| 79 | \( 1 - 2.08T + 79T^{2} \) |
| 83 | \( 1 - 2.93T + 83T^{2} \) |
| 89 | \( 1 - 9.36T + 89T^{2} \) |
| 97 | \( 1 + 4.28T + 97T^{2} \) |
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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
−7.956707461280951562752534691886, −7.31228092231020155921883598999, −6.22357950234128454484308587073, −5.67872740642669887512403977592, −5.23790072778816289205968575686, −4.36986885231356884318029716358, −3.34702070649332643370121607842, −2.45528500180910357669770665482, −1.99701771999799088221681919651, −0.918149816013235111575861333235,
0.918149816013235111575861333235, 1.99701771999799088221681919651, 2.45528500180910357669770665482, 3.34702070649332643370121607842, 4.36986885231356884318029716358, 5.23790072778816289205968575686, 5.67872740642669887512403977592, 6.22357950234128454484308587073, 7.31228092231020155921883598999, 7.956707461280951562752534691886