| L(s) = 1 | − 0.414·3-s − 3.41·5-s − 2.82·9-s − 11-s + 1.82·13-s + 1.41·15-s − 7.65·17-s + 3.41·19-s − 2.24·23-s + 6.65·25-s + 2.41·27-s − 8.65·29-s + 4·31-s + 0.414·33-s − 6.58·37-s − 0.757·39-s − 2.58·41-s − 5.65·43-s + 9.65·45-s − 6.48·47-s + 3.17·51-s − 11.8·53-s + 3.41·55-s − 1.41·57-s + 8.41·59-s + 6.17·61-s − 6.24·65-s + ⋯ |
| L(s) = 1 | − 0.239·3-s − 1.52·5-s − 0.942·9-s − 0.301·11-s + 0.507·13-s + 0.365·15-s − 1.85·17-s + 0.783·19-s − 0.467·23-s + 1.33·25-s + 0.464·27-s − 1.60·29-s + 0.718·31-s + 0.0721·33-s − 1.08·37-s − 0.121·39-s − 0.403·41-s − 0.862·43-s + 1.43·45-s − 0.945·47-s + 0.444·51-s − 1.63·53-s + 0.460·55-s − 0.187·57-s + 1.09·59-s + 0.790·61-s − 0.774·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2273303512\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2273303512\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + 0.414T + 3T^{2} \) |
| 5 | \( 1 + 3.41T + 5T^{2} \) |
| 13 | \( 1 - 1.82T + 13T^{2} \) |
| 17 | \( 1 + 7.65T + 17T^{2} \) |
| 19 | \( 1 - 3.41T + 19T^{2} \) |
| 23 | \( 1 + 2.24T + 23T^{2} \) |
| 29 | \( 1 + 8.65T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 6.58T + 37T^{2} \) |
| 41 | \( 1 + 2.58T + 41T^{2} \) |
| 43 | \( 1 + 5.65T + 43T^{2} \) |
| 47 | \( 1 + 6.48T + 47T^{2} \) |
| 53 | \( 1 + 11.8T + 53T^{2} \) |
| 59 | \( 1 - 8.41T + 59T^{2} \) |
| 61 | \( 1 - 6.17T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 + 3.07T + 71T^{2} \) |
| 73 | \( 1 + 6.58T + 73T^{2} \) |
| 79 | \( 1 - 4.75T + 79T^{2} \) |
| 83 | \( 1 + 16.1T + 83T^{2} \) |
| 89 | \( 1 + 4.48T + 89T^{2} \) |
| 97 | \( 1 - 1.82T + 97T^{2} \) |
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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
−7.86235093057928049345003915773, −7.08172930204316957374182286745, −6.52995438067487670437274869847, −5.65874149684560381708502473700, −4.92830390420495265771885251492, −4.21788074052587846924897411712, −3.51100906737367182630481271256, −2.85495645664312258994843504940, −1.71007699948514299823456004611, −0.22692525678679928252089930112,
0.22692525678679928252089930112, 1.71007699948514299823456004611, 2.85495645664312258994843504940, 3.51100906737367182630481271256, 4.21788074052587846924897411712, 4.92830390420495265771885251492, 5.65874149684560381708502473700, 6.52995438067487670437274869847, 7.08172930204316957374182286745, 7.86235093057928049345003915773
