| L(s) = 1 | − 2.74·3-s − 4.14·5-s − 3.66·7-s + 4.51·9-s + 2.70·11-s + 4.94·13-s + 11.3·15-s − 1.67·17-s + 6.88·19-s + 10.0·21-s + 4.06·23-s + 12.1·25-s − 4.15·27-s − 6.03·29-s − 0.559·31-s − 7.41·33-s + 15.1·35-s − 1.38·37-s − 13.5·39-s − 7.23·41-s + 2.36·43-s − 18.7·45-s + 9.14·47-s + 6.41·49-s + 4.59·51-s + 11.6·53-s − 11.2·55-s + ⋯ |
| L(s) = 1 | − 1.58·3-s − 1.85·5-s − 1.38·7-s + 1.50·9-s + 0.815·11-s + 1.37·13-s + 2.93·15-s − 0.406·17-s + 1.57·19-s + 2.19·21-s + 0.848·23-s + 2.43·25-s − 0.798·27-s − 1.12·29-s − 0.100·31-s − 1.29·33-s + 2.56·35-s − 0.227·37-s − 2.17·39-s − 1.13·41-s + 0.360·43-s − 2.79·45-s + 1.33·47-s + 0.915·49-s + 0.643·51-s + 1.60·53-s − 1.51·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5860232949\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5860232949\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 + 2.74T + 3T^{2} \) |
| 5 | \( 1 + 4.14T + 5T^{2} \) |
| 7 | \( 1 + 3.66T + 7T^{2} \) |
| 11 | \( 1 - 2.70T + 11T^{2} \) |
| 13 | \( 1 - 4.94T + 13T^{2} \) |
| 17 | \( 1 + 1.67T + 17T^{2} \) |
| 19 | \( 1 - 6.88T + 19T^{2} \) |
| 23 | \( 1 - 4.06T + 23T^{2} \) |
| 29 | \( 1 + 6.03T + 29T^{2} \) |
| 31 | \( 1 + 0.559T + 31T^{2} \) |
| 37 | \( 1 + 1.38T + 37T^{2} \) |
| 41 | \( 1 + 7.23T + 41T^{2} \) |
| 43 | \( 1 - 2.36T + 43T^{2} \) |
| 47 | \( 1 - 9.14T + 47T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 - 7.42T + 59T^{2} \) |
| 61 | \( 1 - 1.42T + 61T^{2} \) |
| 67 | \( 1 - 7.90T + 67T^{2} \) |
| 71 | \( 1 - 1.27T + 71T^{2} \) |
| 73 | \( 1 + 12.5T + 73T^{2} \) |
| 79 | \( 1 + 12.5T + 79T^{2} \) |
| 83 | \( 1 + 5.14T + 83T^{2} \) |
| 89 | \( 1 - 8.12T + 89T^{2} \) |
| 97 | \( 1 + 6.29T + 97T^{2} \) |
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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
−7.40593831262631286002742286084, −7.10310605977074514376027551167, −6.51822102362272948596360549128, −5.78963279596369472070511394002, −5.14571154371381474845588602434, −4.10620518012200734237172001229, −3.74139980213275092121464619016, −3.08124260712721906467696253526, −1.14089630388203410503788346797, −0.50721675037494475540129709615,
0.50721675037494475540129709615, 1.14089630388203410503788346797, 3.08124260712721906467696253526, 3.74139980213275092121464619016, 4.10620518012200734237172001229, 5.14571154371381474845588602434, 5.78963279596369472070511394002, 6.51822102362272948596360549128, 7.10310605977074514376027551167, 7.40593831262631286002742286084
