| L(s) = 1 | + 2.41·2-s + 3.82·4-s + 3.41·5-s + 0.828·7-s + 4.41·8-s + 8.24·10-s − 11-s − 13-s + 1.99·14-s + 2.99·16-s + 2.58·17-s − 6·19-s + 13.0·20-s − 2.41·22-s − 4.82·23-s + 6.65·25-s − 2.41·26-s + 3.17·28-s + 4.24·29-s − 4.24·31-s − 1.58·32-s + 6.24·34-s + 2.82·35-s − 3.65·37-s − 14.4·38-s + 15.0·40-s + 12·41-s + ⋯ |
| L(s) = 1 | + 1.70·2-s + 1.91·4-s + 1.52·5-s + 0.313·7-s + 1.56·8-s + 2.60·10-s − 0.301·11-s − 0.277·13-s + 0.534·14-s + 0.749·16-s + 0.627·17-s − 1.37·19-s + 2.92·20-s − 0.514·22-s − 1.00·23-s + 1.33·25-s − 0.473·26-s + 0.599·28-s + 0.787·29-s − 0.762·31-s − 0.280·32-s + 1.07·34-s + 0.478·35-s − 0.601·37-s − 2.34·38-s + 2.38·40-s + 1.87·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.558216059\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.558216059\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 2.41T + 2T^{2} \) |
| 5 | \( 1 - 3.41T + 5T^{2} \) |
| 7 | \( 1 - 0.828T + 7T^{2} \) |
| 17 | \( 1 - 2.58T + 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 + 4.82T + 23T^{2} \) |
| 29 | \( 1 - 4.24T + 29T^{2} \) |
| 31 | \( 1 + 4.24T + 31T^{2} \) |
| 37 | \( 1 + 3.65T + 37T^{2} \) |
| 41 | \( 1 - 12T + 41T^{2} \) |
| 43 | \( 1 + 9.07T + 43T^{2} \) |
| 47 | \( 1 + 8.48T + 47T^{2} \) |
| 53 | \( 1 - 9.31T + 53T^{2} \) |
| 59 | \( 1 - 5.17T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 11.0T + 67T^{2} \) |
| 71 | \( 1 + 5.65T + 71T^{2} \) |
| 73 | \( 1 + 4.48T + 73T^{2} \) |
| 79 | \( 1 - 5.07T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 - 12.5T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−9.972943451852509082524214690159, −8.897534972824754653844045061365, −7.81670996958978331703080215918, −6.65037286860530378352900910304, −6.13342030936368979702987114947, −5.37154642318180008033680917108, −4.73775209357028540215389142677, −3.66109520000920906804419843151, −2.48647326540155919468286844903, −1.84358501593279562301464740504,
1.84358501593279562301464740504, 2.48647326540155919468286844903, 3.66109520000920906804419843151, 4.73775209357028540215389142677, 5.37154642318180008033680917108, 6.13342030936368979702987114947, 6.65037286860530378352900910304, 7.81670996958978331703080215918, 8.897534972824754653844045061365, 9.972943451852509082524214690159
