| L(s) = 1 | + 2.66·3-s − 1.66·5-s + 0.545·7-s + 4.12·9-s + 4.21·11-s − 0.123·13-s − 4.45·15-s − 0.909·17-s + 2·19-s + 1.45·21-s + 4.12·23-s − 2.21·25-s + 3.00·27-s − 1.21·29-s + 5.91·31-s + 11.2·33-s − 0.909·35-s + 37-s − 0.330·39-s + 1.33·41-s + 10.2·43-s − 6.88·45-s − 2.54·47-s − 6.70·49-s − 2.42·51-s + 6.79·53-s − 7.03·55-s + ⋯ |
| L(s) = 1 | + 1.54·3-s − 0.746·5-s + 0.206·7-s + 1.37·9-s + 1.27·11-s − 0.0343·13-s − 1.15·15-s − 0.220·17-s + 0.458·19-s + 0.317·21-s + 0.859·23-s − 0.442·25-s + 0.577·27-s − 0.225·29-s + 1.06·31-s + 1.95·33-s − 0.153·35-s + 0.164·37-s − 0.0529·39-s + 0.207·41-s + 1.56·43-s − 1.02·45-s − 0.371·47-s − 0.957·49-s − 0.340·51-s + 0.933·53-s − 0.948·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.110103401\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.110103401\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 - 2.66T + 3T^{2} \) |
| 5 | \( 1 + 1.66T + 5T^{2} \) |
| 7 | \( 1 - 0.545T + 7T^{2} \) |
| 11 | \( 1 - 4.21T + 11T^{2} \) |
| 13 | \( 1 + 0.123T + 13T^{2} \) |
| 17 | \( 1 + 0.909T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 4.12T + 23T^{2} \) |
| 29 | \( 1 + 1.21T + 29T^{2} \) |
| 31 | \( 1 - 5.91T + 31T^{2} \) |
| 41 | \( 1 - 1.33T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 + 2.54T + 47T^{2} \) |
| 53 | \( 1 - 6.79T + 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 - 2.75T + 61T^{2} \) |
| 67 | \( 1 - 2.33T + 67T^{2} \) |
| 71 | \( 1 + 7.88T + 71T^{2} \) |
| 73 | \( 1 + 13.5T + 73T^{2} \) |
| 79 | \( 1 + 2.78T + 79T^{2} \) |
| 83 | \( 1 + 5.88T + 83T^{2} \) |
| 89 | \( 1 - 9.09T + 89T^{2} \) |
| 97 | \( 1 - 3.81T + 97T^{2} \) |
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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
−8.853404159519842798617226973738, −8.339093109364638708634270914581, −7.52471118694017363201547817585, −7.02900808469001784560683264523, −5.95965400452482072426200028482, −4.62254792482686252070544912611, −3.95313122514777065644576783247, −3.26945500470262064779265008507, −2.32689601095552572914703495251, −1.14605504805363220264162246605,
1.14605504805363220264162246605, 2.32689601095552572914703495251, 3.26945500470262064779265008507, 3.95313122514777065644576783247, 4.62254792482686252070544912611, 5.95965400452482072426200028482, 7.02900808469001784560683264523, 7.52471118694017363201547817585, 8.339093109364638708634270914581, 8.853404159519842798617226973738
