| L(s) = 1 | − 2-s + 4-s − 1.31·5-s − 2.35·7-s − 8-s + 1.31·10-s + 5.16·11-s − 3.08·13-s + 2.35·14-s + 16-s − 4.45·17-s − 0.325·19-s − 1.31·20-s − 5.16·22-s − 5.93·23-s − 3.26·25-s + 3.08·26-s − 2.35·28-s − 5.26·29-s + 2.01·31-s − 32-s + 4.45·34-s + 3.09·35-s − 3.71·37-s + 0.325·38-s + 1.31·40-s + 2.23·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.589·5-s − 0.888·7-s − 0.353·8-s + 0.416·10-s + 1.55·11-s − 0.854·13-s + 0.628·14-s + 0.250·16-s − 1.07·17-s − 0.0747·19-s − 0.294·20-s − 1.10·22-s − 1.23·23-s − 0.652·25-s + 0.604·26-s − 0.444·28-s − 0.977·29-s + 0.361·31-s − 0.176·32-s + 0.763·34-s + 0.523·35-s − 0.611·37-s + 0.0528·38-s + 0.208·40-s + 0.348·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5777238452\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5777238452\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 5 | \( 1 + 1.31T + 5T^{2} \) |
| 7 | \( 1 + 2.35T + 7T^{2} \) |
| 11 | \( 1 - 5.16T + 11T^{2} \) |
| 13 | \( 1 + 3.08T + 13T^{2} \) |
| 17 | \( 1 + 4.45T + 17T^{2} \) |
| 19 | \( 1 + 0.325T + 19T^{2} \) |
| 23 | \( 1 + 5.93T + 23T^{2} \) |
| 29 | \( 1 + 5.26T + 29T^{2} \) |
| 31 | \( 1 - 2.01T + 31T^{2} \) |
| 37 | \( 1 + 3.71T + 37T^{2} \) |
| 41 | \( 1 - 2.23T + 41T^{2} \) |
| 43 | \( 1 - 9.22T + 43T^{2} \) |
| 47 | \( 1 + 6.42T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 - 8.60T + 59T^{2} \) |
| 61 | \( 1 + 0.0686T + 61T^{2} \) |
| 67 | \( 1 - 13.8T + 67T^{2} \) |
| 71 | \( 1 - 12.0T + 71T^{2} \) |
| 73 | \( 1 + 11.5T + 73T^{2} \) |
| 79 | \( 1 - 11.9T + 79T^{2} \) |
| 83 | \( 1 + 1.79T + 83T^{2} \) |
| 89 | \( 1 - 3.84T + 89T^{2} \) |
| 97 | \( 1 - 2.71T + 97T^{2} \) |
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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
−7.82157549973713745612381717773, −7.20231617856675132871880087484, −6.44727663556237939601452694501, −6.18953028927422577689955827414, −5.00501370066877191010065593313, −3.97913705961454780643221419560, −3.67602888295962027304767522640, −2.50832117712608338227752150015, −1.72826926781117542717642162382, −0.40597830867825580303218179224,
0.40597830867825580303218179224, 1.72826926781117542717642162382, 2.50832117712608338227752150015, 3.67602888295962027304767522640, 3.97913705961454780643221419560, 5.00501370066877191010065593313, 6.18953028927422577689955827414, 6.44727663556237939601452694501, 7.20231617856675132871880087484, 7.82157549973713745612381717773
