| L(s) = 1 | − 0.776·2-s + 3.01·3-s − 1.39·4-s − 0.0164·5-s − 2.33·6-s − 0.793·7-s + 2.63·8-s + 6.06·9-s + 0.0127·10-s − 11-s − 4.20·12-s + 0.616·14-s − 0.0495·15-s + 0.747·16-s + 3.12·17-s − 4.70·18-s − 1.22·19-s + 0.0229·20-s − 2.38·21-s + 0.776·22-s + 4.26·23-s + 7.93·24-s − 4.99·25-s + 9.21·27-s + 1.10·28-s + 8.01·29-s + 0.0384·30-s + ⋯ |
| L(s) = 1 | − 0.548·2-s + 1.73·3-s − 0.698·4-s − 0.00735·5-s − 0.953·6-s − 0.299·7-s + 0.932·8-s + 2.02·9-s + 0.00403·10-s − 0.301·11-s − 1.21·12-s + 0.164·14-s − 0.0127·15-s + 0.186·16-s + 0.758·17-s − 1.10·18-s − 0.281·19-s + 0.00514·20-s − 0.521·21-s + 0.165·22-s + 0.890·23-s + 1.62·24-s − 0.999·25-s + 1.77·27-s + 0.209·28-s + 1.48·29-s + 0.00702·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.091542805\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.091542805\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 0.776T + 2T^{2} \) |
| 3 | \( 1 - 3.01T + 3T^{2} \) |
| 5 | \( 1 + 0.0164T + 5T^{2} \) |
| 7 | \( 1 + 0.793T + 7T^{2} \) |
| 17 | \( 1 - 3.12T + 17T^{2} \) |
| 19 | \( 1 + 1.22T + 19T^{2} \) |
| 23 | \( 1 - 4.26T + 23T^{2} \) |
| 29 | \( 1 - 8.01T + 29T^{2} \) |
| 31 | \( 1 + 2.49T + 31T^{2} \) |
| 37 | \( 1 - 8.88T + 37T^{2} \) |
| 41 | \( 1 - 8.43T + 41T^{2} \) |
| 43 | \( 1 - 11.8T + 43T^{2} \) |
| 47 | \( 1 + 12.2T + 47T^{2} \) |
| 53 | \( 1 - 4.56T + 53T^{2} \) |
| 59 | \( 1 + 0.265T + 59T^{2} \) |
| 61 | \( 1 + 4.49T + 61T^{2} \) |
| 67 | \( 1 + 5.31T + 67T^{2} \) |
| 71 | \( 1 - 13.1T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 + 9.35T + 79T^{2} \) |
| 83 | \( 1 - 4.81T + 83T^{2} \) |
| 89 | \( 1 - 1.66T + 89T^{2} \) |
| 97 | \( 1 + 8.35T + 97T^{2} \) |
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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
−9.227356833363515596045933276796, −8.502626826863905738482539098381, −7.84046354248518948446845093759, −7.44797875457810097477987842510, −6.19775712498467604773597387346, −4.89699993907720020788983686784, −4.08170550512725666942254944895, −3.24454634110249998393006527189, −2.33682064660647311484667660059, −1.04602453798382507937057180977,
1.04602453798382507937057180977, 2.33682064660647311484667660059, 3.24454634110249998393006527189, 4.08170550512725666942254944895, 4.89699993907720020788983686784, 6.19775712498467604773597387346, 7.44797875457810097477987842510, 7.84046354248518948446845093759, 8.502626826863905738482539098381, 9.227356833363515596045933276796
