| L(s) = 1 | + 2.34·3-s + 5-s − 3.96·7-s + 2.48·9-s − 1.61·11-s + 3.77·13-s + 2.34·15-s − 4.26·17-s − 5.74·19-s − 9.27·21-s + 8.20·23-s + 25-s − 1.20·27-s + 2.71·29-s − 2.56·31-s − 3.77·33-s − 3.96·35-s − 9.27·37-s + 8.83·39-s + 9.30·41-s − 6.10·43-s + 2.48·45-s + 11.1·47-s + 8.68·49-s − 9.97·51-s − 10.8·53-s − 1.61·55-s + ⋯ |
| L(s) = 1 | + 1.35·3-s + 0.447·5-s − 1.49·7-s + 0.828·9-s − 0.485·11-s + 1.04·13-s + 0.604·15-s − 1.03·17-s − 1.31·19-s − 2.02·21-s + 1.71·23-s + 0.200·25-s − 0.232·27-s + 0.503·29-s − 0.460·31-s − 0.656·33-s − 0.669·35-s − 1.52·37-s + 1.41·39-s + 1.45·41-s − 0.930·43-s + 0.370·45-s + 1.62·47-s + 1.24·49-s − 1.39·51-s − 1.48·53-s − 0.217·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 3 | \( 1 - 2.34T + 3T^{2} \) |
| 7 | \( 1 + 3.96T + 7T^{2} \) |
| 11 | \( 1 + 1.61T + 11T^{2} \) |
| 13 | \( 1 - 3.77T + 13T^{2} \) |
| 17 | \( 1 + 4.26T + 17T^{2} \) |
| 19 | \( 1 + 5.74T + 19T^{2} \) |
| 23 | \( 1 - 8.20T + 23T^{2} \) |
| 29 | \( 1 - 2.71T + 29T^{2} \) |
| 31 | \( 1 + 2.56T + 31T^{2} \) |
| 37 | \( 1 + 9.27T + 37T^{2} \) |
| 41 | \( 1 - 9.30T + 41T^{2} \) |
| 43 | \( 1 + 6.10T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 - 3.71T + 59T^{2} \) |
| 61 | \( 1 + 2.03T + 61T^{2} \) |
| 67 | \( 1 + 8.76T + 67T^{2} \) |
| 71 | \( 1 - 0.0492T + 71T^{2} \) |
| 73 | \( 1 + 15.9T + 73T^{2} \) |
| 79 | \( 1 + 8.62T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 + 2.54T + 89T^{2} \) |
| 97 | \( 1 + 6.49T + 97T^{2} \) |
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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
−7.47540980404437022211785815180, −6.77743205864042041410664963128, −6.30630031341883103129983001960, −5.50346862853292689235023950468, −4.40750227851039864091378831798, −3.70558353387050171631133808763, −2.94104125175022969672591174312, −2.55035274391452135375880371271, −1.50665321566694773709929728739, 0,
1.50665321566694773709929728739, 2.55035274391452135375880371271, 2.94104125175022969672591174312, 3.70558353387050171631133808763, 4.40750227851039864091378831798, 5.50346862853292689235023950468, 6.30630031341883103129983001960, 6.77743205864042041410664963128, 7.47540980404437022211785815180
