| L(s) = 1 | − 2.75·2-s − 1.09·3-s + 5.60·4-s + 2.84·5-s + 3.02·6-s + 3.79·7-s − 9.93·8-s − 1.79·9-s − 7.85·10-s − 0.397·11-s − 6.15·12-s − 2.48·13-s − 10.4·14-s − 3.12·15-s + 16.1·16-s + 2.84·17-s + 4.94·18-s − 2.45·19-s + 15.9·20-s − 4.16·21-s + 1.09·22-s + 3.57·23-s + 10.9·24-s + 3.12·25-s + 6.84·26-s + 5.26·27-s + 21.2·28-s + ⋯ |
| L(s) = 1 | − 1.94·2-s − 0.633·3-s + 2.80·4-s + 1.27·5-s + 1.23·6-s + 1.43·7-s − 3.51·8-s − 0.598·9-s − 2.48·10-s − 0.119·11-s − 1.77·12-s − 0.688·13-s − 2.79·14-s − 0.807·15-s + 4.04·16-s + 0.690·17-s + 1.16·18-s − 0.563·19-s + 3.56·20-s − 0.909·21-s + 0.233·22-s + 0.746·23-s + 2.22·24-s + 0.624·25-s + 1.34·26-s + 1.01·27-s + 4.01·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8747017552\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8747017552\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 6037 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.75T + 2T^{2} \) |
| 3 | \( 1 + 1.09T + 3T^{2} \) |
| 5 | \( 1 - 2.84T + 5T^{2} \) |
| 7 | \( 1 - 3.79T + 7T^{2} \) |
| 11 | \( 1 + 0.397T + 11T^{2} \) |
| 13 | \( 1 + 2.48T + 13T^{2} \) |
| 17 | \( 1 - 2.84T + 17T^{2} \) |
| 19 | \( 1 + 2.45T + 19T^{2} \) |
| 23 | \( 1 - 3.57T + 23T^{2} \) |
| 29 | \( 1 - 1.79T + 29T^{2} \) |
| 31 | \( 1 + 0.342T + 31T^{2} \) |
| 37 | \( 1 + 6.20T + 37T^{2} \) |
| 41 | \( 1 - 1.27T + 41T^{2} \) |
| 43 | \( 1 + 3.33T + 43T^{2} \) |
| 47 | \( 1 + 0.761T + 47T^{2} \) |
| 53 | \( 1 - 3.51T + 53T^{2} \) |
| 59 | \( 1 + 0.993T + 59T^{2} \) |
| 61 | \( 1 - 6.36T + 61T^{2} \) |
| 67 | \( 1 + 4.46T + 67T^{2} \) |
| 71 | \( 1 - 4.45T + 71T^{2} \) |
| 73 | \( 1 - 0.508T + 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 + 2.24T + 83T^{2} \) |
| 89 | \( 1 - 5.76T + 89T^{2} \) |
| 97 | \( 1 - 0.292T + 97T^{2} \) |
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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
−8.331487549782252453736603402175, −7.48909000582042852033669432410, −6.87786762445554619768700195520, −6.06599695727125161193805804689, −5.51303746898011012118501792861, −4.86843101448285379745419917495, −3.07653943136299376661906838673, −2.20852796762826830868863654525, −1.64734032892183332214920807713, −0.69455666692865456117035872013,
0.69455666692865456117035872013, 1.64734032892183332214920807713, 2.20852796762826830868863654525, 3.07653943136299376661906838673, 4.86843101448285379745419917495, 5.51303746898011012118501792861, 6.06599695727125161193805804689, 6.87786762445554619768700195520, 7.48909000582042852033669432410, 8.331487549782252453736603402175
