| L(s) = 1 | − 2.56·2-s + 2.69·3-s + 4.58·4-s + 1.90·5-s − 6.90·6-s − 1.37·7-s − 6.64·8-s + 4.24·9-s − 4.88·10-s − 2.14·11-s + 12.3·12-s + 6.04·13-s + 3.51·14-s + 5.11·15-s + 7.87·16-s + 4.83·17-s − 10.8·18-s − 2.20·19-s + 8.72·20-s − 3.68·21-s + 5.50·22-s − 0.879·23-s − 17.8·24-s − 1.38·25-s − 15.5·26-s + 3.34·27-s − 6.29·28-s + ⋯ |
| L(s) = 1 | − 1.81·2-s + 1.55·3-s + 2.29·4-s + 0.850·5-s − 2.82·6-s − 0.518·7-s − 2.34·8-s + 1.41·9-s − 1.54·10-s − 0.646·11-s + 3.56·12-s + 1.67·13-s + 0.940·14-s + 1.32·15-s + 1.96·16-s + 1.17·17-s − 2.56·18-s − 0.505·19-s + 1.95·20-s − 0.805·21-s + 1.17·22-s − 0.183·23-s − 3.65·24-s − 0.276·25-s − 3.04·26-s + 0.643·27-s − 1.18·28-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
| \(L(1)\) |
\(\approx\) |
\(1.941189242\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.941189242\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.56T + 2T^{2} \) |
| 3 | \( 1 - 2.69T + 3T^{2} \) |
| 5 | \( 1 - 1.90T + 5T^{2} \) |
| 7 | \( 1 + 1.37T + 7T^{2} \) |
| 11 | \( 1 + 2.14T + 11T^{2} \) |
| 13 | \( 1 - 6.04T + 13T^{2} \) |
| 17 | \( 1 - 4.83T + 17T^{2} \) |
| 19 | \( 1 + 2.20T + 19T^{2} \) |
| 23 | \( 1 + 0.879T + 23T^{2} \) |
| 29 | \( 1 - 2.12T + 29T^{2} \) |
| 31 | \( 1 - 7.97T + 31T^{2} \) |
| 37 | \( 1 + 1.36T + 37T^{2} \) |
| 41 | \( 1 - 2.73T + 41T^{2} \) |
| 43 | \( 1 - 1.23T + 43T^{2} \) |
| 47 | \( 1 - 8.84T + 47T^{2} \) |
| 53 | \( 1 + 6.94T + 53T^{2} \) |
| 59 | \( 1 - 14.3T + 59T^{2} \) |
| 61 | \( 1 - 8.87T + 61T^{2} \) |
| 67 | \( 1 + 8.94T + 67T^{2} \) |
| 71 | \( 1 - 3.40T + 71T^{2} \) |
| 73 | \( 1 + 0.293T + 73T^{2} \) |
| 79 | \( 1 + 0.569T + 79T^{2} \) |
| 83 | \( 1 - 0.837T + 83T^{2} \) |
| 89 | \( 1 + 4.33T + 89T^{2} \) |
| 97 | \( 1 + 2.53T + 97T^{2} \) |
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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
−8.168700678312649523471533969881, −7.894691690007592593899277442966, −6.94707532077152114966596916048, −6.27071185108375280708854789736, −5.62325887054791848711328632788, −4.02017755207125981559440249488, −3.13060790000868814376815028248, −2.54850900818914820087978033700, −1.75769795106928278847031410592, −0.929536098906663667154782681546,
0.929536098906663667154782681546, 1.75769795106928278847031410592, 2.54850900818914820087978033700, 3.13060790000868814376815028248, 4.02017755207125981559440249488, 5.62325887054791848711328632788, 6.27071185108375280708854789736, 6.94707532077152114966596916048, 7.894691690007592593899277442966, 8.168700678312649523471533969881
