| L(s) = 1 | − 2.97·3-s + 2.62·5-s + 3.05·7-s + 5.84·9-s + 6.02·11-s − 0.344·13-s − 7.82·15-s − 7.07·17-s + 1.71·19-s − 9.07·21-s − 4.76·23-s + 1.91·25-s − 8.46·27-s + 9.82·29-s + 0.127·31-s − 17.9·33-s + 8.02·35-s + 5.66·37-s + 1.02·39-s + 4.50·41-s − 6.66·43-s + 15.3·45-s + 4.84·47-s + 2.31·49-s + 21.0·51-s + 2.84·53-s + 15.8·55-s + ⋯ |
| L(s) = 1 | − 1.71·3-s + 1.17·5-s + 1.15·7-s + 1.94·9-s + 1.81·11-s − 0.0956·13-s − 2.01·15-s − 1.71·17-s + 0.393·19-s − 1.98·21-s − 0.994·23-s + 0.382·25-s − 1.62·27-s + 1.82·29-s + 0.0229·31-s − 3.12·33-s + 1.35·35-s + 0.931·37-s + 0.164·39-s + 0.703·41-s − 1.01·43-s + 2.29·45-s + 0.706·47-s + 0.331·49-s + 2.94·51-s + 0.391·53-s + 2.13·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 976 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 976 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.386367181\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.386367181\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 61 | \( 1 + T \) |
| good | 3 | \( 1 + 2.97T + 3T^{2} \) |
| 5 | \( 1 - 2.62T + 5T^{2} \) |
| 7 | \( 1 - 3.05T + 7T^{2} \) |
| 11 | \( 1 - 6.02T + 11T^{2} \) |
| 13 | \( 1 + 0.344T + 13T^{2} \) |
| 17 | \( 1 + 7.07T + 17T^{2} \) |
| 19 | \( 1 - 1.71T + 19T^{2} \) |
| 23 | \( 1 + 4.76T + 23T^{2} \) |
| 29 | \( 1 - 9.82T + 29T^{2} \) |
| 31 | \( 1 - 0.127T + 31T^{2} \) |
| 37 | \( 1 - 5.66T + 37T^{2} \) |
| 41 | \( 1 - 4.50T + 41T^{2} \) |
| 43 | \( 1 + 6.66T + 43T^{2} \) |
| 47 | \( 1 - 4.84T + 47T^{2} \) |
| 53 | \( 1 - 2.84T + 53T^{2} \) |
| 59 | \( 1 + 4.31T + 59T^{2} \) |
| 67 | \( 1 - 13.0T + 67T^{2} \) |
| 71 | \( 1 + 4.44T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 - 7.12T + 79T^{2} \) |
| 83 | \( 1 + 9.69T + 83T^{2} \) |
| 89 | \( 1 + 4.67T + 89T^{2} \) |
| 97 | \( 1 - 8.51T + 97T^{2} \) |
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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
−10.14570947598313369077796796834, −9.388067630377039945476189586843, −8.455529578366395912409642866122, −7.05642213202246236785710586985, −6.34434039786369212408953120431, −5.88187670247273219267157644418, −4.77103022400674699085576756493, −4.28358979838844296969975450029, −2.03760015817707503539696098249, −1.11115472466280201906502364703,
1.11115472466280201906502364703, 2.03760015817707503539696098249, 4.28358979838844296969975450029, 4.77103022400674699085576756493, 5.88187670247273219267157644418, 6.34434039786369212408953120431, 7.05642213202246236785710586985, 8.455529578366395912409642866122, 9.388067630377039945476189586843, 10.14570947598313369077796796834
