| L(s) = 1 | + 2.71·2-s + 5.37·4-s + 1.58·5-s + 9.15·8-s + 4.30·10-s + 1.34·11-s − 3.17·13-s + 14.1·16-s − 2.80·17-s + 0.625·19-s + 8.52·20-s + 3.66·22-s + 0.284·23-s − 2.48·25-s − 8.62·26-s + 4.54·29-s + 7.43·31-s + 20.0·32-s − 7.60·34-s + 8.02·37-s + 1.69·38-s + 14.5·40-s + 10.0·41-s + 6.25·43-s + 7.24·44-s + 0.773·46-s − 11.1·47-s + ⋯ |
| L(s) = 1 | + 1.91·2-s + 2.68·4-s + 0.709·5-s + 3.23·8-s + 1.36·10-s + 0.406·11-s − 0.881·13-s + 3.52·16-s − 0.679·17-s + 0.143·19-s + 1.90·20-s + 0.780·22-s + 0.0593·23-s − 0.496·25-s − 1.69·26-s + 0.844·29-s + 1.33·31-s + 3.53·32-s − 1.30·34-s + 1.31·37-s + 0.275·38-s + 2.29·40-s + 1.56·41-s + 0.954·43-s + 1.09·44-s + 0.114·46-s − 1.62·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.026925107\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.026925107\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 2.71T + 2T^{2} \) |
| 5 | \( 1 - 1.58T + 5T^{2} \) |
| 11 | \( 1 - 1.34T + 11T^{2} \) |
| 13 | \( 1 + 3.17T + 13T^{2} \) |
| 17 | \( 1 + 2.80T + 17T^{2} \) |
| 19 | \( 1 - 0.625T + 19T^{2} \) |
| 23 | \( 1 - 0.284T + 23T^{2} \) |
| 29 | \( 1 - 4.54T + 29T^{2} \) |
| 31 | \( 1 - 7.43T + 31T^{2} \) |
| 37 | \( 1 - 8.02T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 - 6.25T + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 + 2.78T + 53T^{2} \) |
| 59 | \( 1 + 4.57T + 59T^{2} \) |
| 61 | \( 1 + 0.385T + 61T^{2} \) |
| 67 | \( 1 + 2.53T + 67T^{2} \) |
| 71 | \( 1 - 1.45T + 71T^{2} \) |
| 73 | \( 1 + 0.468T + 73T^{2} \) |
| 79 | \( 1 + 15.7T + 79T^{2} \) |
| 83 | \( 1 + 13.9T + 83T^{2} \) |
| 89 | \( 1 - 2.58T + 89T^{2} \) |
| 97 | \( 1 + 14.4T + 97T^{2} \) |
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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
−8.143221270748261512233850245926, −7.39631969420960813072338970671, −6.57773569959106832646379659215, −6.13296170359448342729398997681, −5.42460581160310528648352562259, −4.53698471649788608868643104582, −4.19923212709903729486750053933, −2.89554627032715163721987897551, −2.48721201372077896595122162366, −1.40762867536498682071059022224,
1.40762867536498682071059022224, 2.48721201372077896595122162366, 2.89554627032715163721987897551, 4.19923212709903729486750053933, 4.53698471649788608868643104582, 5.42460581160310528648352562259, 6.13296170359448342729398997681, 6.57773569959106832646379659215, 7.39631969420960813072338970671, 8.143221270748261512233850245926
