L(s) = 1 | + 1.06·3-s + 0.381·5-s − 7-s − 1.85·9-s − 11-s − 13-s + 0.408·15-s + 7.47·17-s + 1.51·19-s − 1.06·21-s − 3.96·23-s − 4.85·25-s − 5.19·27-s + 2.36·29-s − 3.61·31-s − 1.06·33-s − 0.381·35-s − 4.82·37-s − 1.06·39-s − 2.43·41-s + 12.2·43-s − 0.708·45-s + 8.39·47-s + 49-s + 7.99·51-s − 5.64·53-s − 0.381·55-s + ⋯ |
L(s) = 1 | + 0.617·3-s + 0.170·5-s − 0.377·7-s − 0.618·9-s − 0.301·11-s − 0.277·13-s + 0.105·15-s + 1.81·17-s + 0.347·19-s − 0.233·21-s − 0.827·23-s − 0.970·25-s − 0.999·27-s + 0.438·29-s − 0.650·31-s − 0.186·33-s − 0.0645·35-s − 0.793·37-s − 0.171·39-s − 0.380·41-s + 1.87·43-s − 0.105·45-s + 1.22·47-s + 0.142·49-s + 1.11·51-s − 0.775·53-s − 0.0514·55-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
\(L(1)\) |
\(\approx\) |
\(2.134973134\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.134973134\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.06T + 3T^{2} \) |
| 5 | \( 1 - 0.381T + 5T^{2} \) |
| 17 | \( 1 - 7.47T + 17T^{2} \) |
| 19 | \( 1 - 1.51T + 19T^{2} \) |
| 23 | \( 1 + 3.96T + 23T^{2} \) |
| 29 | \( 1 - 2.36T + 29T^{2} \) |
| 31 | \( 1 + 3.61T + 31T^{2} \) |
| 37 | \( 1 + 4.82T + 37T^{2} \) |
| 41 | \( 1 + 2.43T + 41T^{2} \) |
| 43 | \( 1 - 12.2T + 43T^{2} \) |
| 47 | \( 1 - 8.39T + 47T^{2} \) |
| 53 | \( 1 + 5.64T + 53T^{2} \) |
| 59 | \( 1 - 7.25T + 59T^{2} \) |
| 61 | \( 1 - 7.70T + 61T^{2} \) |
| 67 | \( 1 - 8.61T + 67T^{2} \) |
| 71 | \( 1 - 6.79T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 - 2.58T + 79T^{2} \) |
| 83 | \( 1 + 1.86T + 83T^{2} \) |
| 89 | \( 1 - 4.79T + 89T^{2} \) |
| 97 | \( 1 - 11.1T + 97T^{2} \) |
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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
−7.88662353683801014290172287444, −7.35550809549356366934059429377, −6.42327689935380364915552200445, −5.50646888760661436664815305240, −5.41316792436703504017095582838, −3.99886456458849833769884566806, −3.49710588833068175066101360836, −2.68834704815619502466939971392, −1.97539117791992037674295431953, −0.68214287450473700532246326923,
0.68214287450473700532246326923, 1.97539117791992037674295431953, 2.68834704815619502466939971392, 3.49710588833068175066101360836, 3.99886456458849833769884566806, 5.41316792436703504017095582838, 5.50646888760661436664815305240, 6.42327689935380364915552200445, 7.35550809549356366934059429377, 7.88662353683801014290172287444