| L(s) = 1 | − 1.18·3-s − 2.06·5-s + 0.279·7-s − 1.60·9-s − 3.49·11-s − 3.50·13-s + 2.43·15-s − 2.93·17-s − 2.42·19-s − 0.330·21-s − 2.39·23-s − 0.743·25-s + 5.43·27-s − 3.15·29-s − 4.46·31-s + 4.12·33-s − 0.577·35-s − 6.97·37-s + 4.13·39-s + 8.78·41-s − 12.9·43-s + 3.31·45-s − 47-s − 6.92·49-s + 3.45·51-s − 4.29·53-s + 7.21·55-s + ⋯ |
| L(s) = 1 | − 0.681·3-s − 0.922·5-s + 0.105·7-s − 0.535·9-s − 1.05·11-s − 0.972·13-s + 0.628·15-s − 0.710·17-s − 0.557·19-s − 0.0720·21-s − 0.498·23-s − 0.148·25-s + 1.04·27-s − 0.586·29-s − 0.801·31-s + 0.718·33-s − 0.0975·35-s − 1.14·37-s + 0.662·39-s + 1.37·41-s − 1.97·43-s + 0.494·45-s − 0.145·47-s − 0.988·49-s + 0.484·51-s − 0.590·53-s + 0.973·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.01573287578\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01573287578\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 47 | \( 1 + T \) |
| good | 3 | \( 1 + 1.18T + 3T^{2} \) |
| 5 | \( 1 + 2.06T + 5T^{2} \) |
| 7 | \( 1 - 0.279T + 7T^{2} \) |
| 11 | \( 1 + 3.49T + 11T^{2} \) |
| 13 | \( 1 + 3.50T + 13T^{2} \) |
| 17 | \( 1 + 2.93T + 17T^{2} \) |
| 19 | \( 1 + 2.42T + 19T^{2} \) |
| 23 | \( 1 + 2.39T + 23T^{2} \) |
| 29 | \( 1 + 3.15T + 29T^{2} \) |
| 31 | \( 1 + 4.46T + 31T^{2} \) |
| 37 | \( 1 + 6.97T + 37T^{2} \) |
| 41 | \( 1 - 8.78T + 41T^{2} \) |
| 43 | \( 1 + 12.9T + 43T^{2} \) |
| 53 | \( 1 + 4.29T + 53T^{2} \) |
| 59 | \( 1 + 1.39T + 59T^{2} \) |
| 61 | \( 1 - 3.74T + 61T^{2} \) |
| 67 | \( 1 + 7.66T + 67T^{2} \) |
| 71 | \( 1 + 1.67T + 71T^{2} \) |
| 73 | \( 1 - 7.21T + 73T^{2} \) |
| 79 | \( 1 - 2.44T + 79T^{2} \) |
| 83 | \( 1 + 14.4T + 83T^{2} \) |
| 89 | \( 1 + 4.63T + 89T^{2} \) |
| 97 | \( 1 - 3.16T + 97T^{2} \) |
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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
−8.037077114357276674628532390021, −7.40832360416439388529231576287, −6.70426543374264293636567874701, −5.86638784240103983224364635094, −5.14628780632994861565766077515, −4.61828223420432733839107864447, −3.69950231322107249418361309280, −2.79895541762780807114149890048, −1.89491064263542506747661055527, −0.06352272992389438252621569200,
0.06352272992389438252621569200, 1.89491064263542506747661055527, 2.79895541762780807114149890048, 3.69950231322107249418361309280, 4.61828223420432733839107864447, 5.14628780632994861565766077515, 5.86638784240103983224364635094, 6.70426543374264293636567874701, 7.40832360416439388529231576287, 8.037077114357276674628532390021
