L(s) = 1 | − 0.430·3-s − 1.65·5-s − 2.73·7-s − 2.81·9-s − 5.21·11-s − 2.94·13-s + 0.711·15-s + 0.219·17-s + 19-s + 1.17·21-s − 3.71·23-s − 2.26·25-s + 2.50·27-s − 4.94·29-s − 3.98·31-s + 2.24·33-s + 4.52·35-s + 2.69·37-s + 1.26·39-s − 4.84·41-s + 1.83·43-s + 4.65·45-s − 6.90·47-s + 0.479·49-s − 0.0947·51-s + 53-s + 8.62·55-s + ⋯ |
L(s) = 1 | − 0.248·3-s − 0.739·5-s − 1.03·7-s − 0.938·9-s − 1.57·11-s − 0.815·13-s + 0.183·15-s + 0.0533·17-s + 0.229·19-s + 0.257·21-s − 0.775·23-s − 0.453·25-s + 0.481·27-s − 0.919·29-s − 0.716·31-s + 0.391·33-s + 0.764·35-s + 0.443·37-s + 0.202·39-s − 0.757·41-s + 0.280·43-s + 0.693·45-s − 1.00·47-s + 0.0685·49-s − 0.0132·51-s + 0.137·53-s + 1.16·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1101795021\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1101795021\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
| 53 | \( 1 - T \) |
good | 3 | \( 1 + 0.430T + 3T^{2} \) |
| 5 | \( 1 + 1.65T + 5T^{2} \) |
| 7 | \( 1 + 2.73T + 7T^{2} \) |
| 11 | \( 1 + 5.21T + 11T^{2} \) |
| 13 | \( 1 + 2.94T + 13T^{2} \) |
| 17 | \( 1 - 0.219T + 17T^{2} \) |
| 23 | \( 1 + 3.71T + 23T^{2} \) |
| 29 | \( 1 + 4.94T + 29T^{2} \) |
| 31 | \( 1 + 3.98T + 31T^{2} \) |
| 37 | \( 1 - 2.69T + 37T^{2} \) |
| 41 | \( 1 + 4.84T + 41T^{2} \) |
| 43 | \( 1 - 1.83T + 43T^{2} \) |
| 47 | \( 1 + 6.90T + 47T^{2} \) |
| 59 | \( 1 - 0.959T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 + 8.29T + 67T^{2} \) |
| 71 | \( 1 - 16.3T + 71T^{2} \) |
| 73 | \( 1 - 2.96T + 73T^{2} \) |
| 79 | \( 1 - 3.81T + 79T^{2} \) |
| 83 | \( 1 - 2.39T + 83T^{2} \) |
| 89 | \( 1 + 4.06T + 89T^{2} \) |
| 97 | \( 1 + 0.432T + 97T^{2} \) |
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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
−8.186886214571374843765720807628, −7.81391705207792545329038699744, −7.07430908848990969505392309851, −6.13511655828125823298407183386, −5.49792279953343270838294440318, −4.80116569907131403950713693208, −3.67707478897183409324859768345, −3.02694350344589490926990274205, −2.18841312683127412273541762020, −0.17520509724943477844285665394,
0.17520509724943477844285665394, 2.18841312683127412273541762020, 3.02694350344589490926990274205, 3.67707478897183409324859768345, 4.80116569907131403950713693208, 5.49792279953343270838294440318, 6.13511655828125823298407183386, 7.07430908848990969505392309851, 7.81391705207792545329038699744, 8.186886214571374843765720807628