| L(s) = 1 | − 3-s + 3.95·5-s + 1.63·7-s + 9-s − 4.82·11-s + 5.59·13-s − 3.95·15-s − 0.828·17-s + 2.82·19-s − 1.63·21-s + 7.91·23-s + 10.6·25-s − 27-s − 7.23·29-s − 1.63·31-s + 4.82·33-s + 6.48·35-s − 2.31·37-s − 5.59·39-s + 3.17·41-s − 4.48·43-s + 3.95·45-s + 7.91·47-s − 4.31·49-s + 0.828·51-s − 0.678·53-s − 19.1·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.76·5-s + 0.619·7-s + 0.333·9-s − 1.45·11-s + 1.55·13-s − 1.02·15-s − 0.200·17-s + 0.648·19-s − 0.357·21-s + 1.65·23-s + 2.13·25-s − 0.192·27-s − 1.34·29-s − 0.294·31-s + 0.840·33-s + 1.09·35-s − 0.381·37-s − 0.896·39-s + 0.495·41-s − 0.683·43-s + 0.589·45-s + 1.15·47-s − 0.616·49-s + 0.116·51-s − 0.0932·53-s − 2.57·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.146272594\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.146272594\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| good | 5 | \( 1 - 3.95T + 5T^{2} \) |
| 7 | \( 1 - 1.63T + 7T^{2} \) |
| 11 | \( 1 + 4.82T + 11T^{2} \) |
| 13 | \( 1 - 5.59T + 13T^{2} \) |
| 17 | \( 1 + 0.828T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 - 7.91T + 23T^{2} \) |
| 29 | \( 1 + 7.23T + 29T^{2} \) |
| 31 | \( 1 + 1.63T + 31T^{2} \) |
| 37 | \( 1 + 2.31T + 37T^{2} \) |
| 41 | \( 1 - 3.17T + 41T^{2} \) |
| 43 | \( 1 + 4.48T + 43T^{2} \) |
| 47 | \( 1 - 7.91T + 47T^{2} \) |
| 53 | \( 1 + 0.678T + 53T^{2} \) |
| 59 | \( 1 + 9.65T + 59T^{2} \) |
| 61 | \( 1 + 2.31T + 61T^{2} \) |
| 67 | \( 1 - 13.6T + 67T^{2} \) |
| 71 | \( 1 + 3.27T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 - 1.63T + 79T^{2} \) |
| 83 | \( 1 + 8.82T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
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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
−9.432716472148106195317556875833, −8.863835811819350820791903661163, −7.83044214549853554762967236770, −6.88336203654565741251071035834, −5.96060594363477265719184694540, −5.42883775999320007188950305115, −4.84641395267085299298420996286, −3.28542176021230469221787547165, −2.12887893026920818926188999439, −1.17914994145276179065768207164,
1.17914994145276179065768207164, 2.12887893026920818926188999439, 3.28542176021230469221787547165, 4.84641395267085299298420996286, 5.42883775999320007188950305115, 5.96060594363477265719184694540, 6.88336203654565741251071035834, 7.83044214549853554762967236770, 8.863835811819350820791903661163, 9.432716472148106195317556875833
