L(s) = 1 | − 2.41·3-s + 0.877·5-s + 2.89·7-s + 2.84·9-s + 1.81·11-s − 3.69·13-s − 2.12·15-s − 5.52·17-s − 3.24·19-s − 6.99·21-s − 2.45·23-s − 4.23·25-s + 0.369·27-s + 2.86·29-s − 7.81·31-s − 4.38·33-s + 2.53·35-s + 4.93·37-s + 8.94·39-s + 12.1·41-s + 6.06·43-s + 2.49·45-s + 4.72·47-s + 1.35·49-s + 13.3·51-s − 1.39·53-s + 1.59·55-s + ⋯ |
L(s) = 1 | − 1.39·3-s + 0.392·5-s + 1.09·7-s + 0.948·9-s + 0.546·11-s − 1.02·13-s − 0.547·15-s − 1.33·17-s − 0.743·19-s − 1.52·21-s − 0.512·23-s − 0.846·25-s + 0.0712·27-s + 0.532·29-s − 1.40·31-s − 0.762·33-s + 0.428·35-s + 0.810·37-s + 1.43·39-s + 1.89·41-s + 0.924·43-s + 0.372·45-s + 0.689·47-s + 0.194·49-s + 1.87·51-s − 0.191·53-s + 0.214·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.060080165\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.060080165\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 751 | \( 1 + T \) |
good | 3 | \( 1 + 2.41T + 3T^{2} \) |
| 5 | \( 1 - 0.877T + 5T^{2} \) |
| 7 | \( 1 - 2.89T + 7T^{2} \) |
| 11 | \( 1 - 1.81T + 11T^{2} \) |
| 13 | \( 1 + 3.69T + 13T^{2} \) |
| 17 | \( 1 + 5.52T + 17T^{2} \) |
| 19 | \( 1 + 3.24T + 19T^{2} \) |
| 23 | \( 1 + 2.45T + 23T^{2} \) |
| 29 | \( 1 - 2.86T + 29T^{2} \) |
| 31 | \( 1 + 7.81T + 31T^{2} \) |
| 37 | \( 1 - 4.93T + 37T^{2} \) |
| 41 | \( 1 - 12.1T + 41T^{2} \) |
| 43 | \( 1 - 6.06T + 43T^{2} \) |
| 47 | \( 1 - 4.72T + 47T^{2} \) |
| 53 | \( 1 + 1.39T + 53T^{2} \) |
| 59 | \( 1 + 8.45T + 59T^{2} \) |
| 61 | \( 1 - 13.2T + 61T^{2} \) |
| 67 | \( 1 + 8.83T + 67T^{2} \) |
| 71 | \( 1 + 3.13T + 71T^{2} \) |
| 73 | \( 1 - 9.61T + 73T^{2} \) |
| 79 | \( 1 - 0.251T + 79T^{2} \) |
| 83 | \( 1 - 1.82T + 83T^{2} \) |
| 89 | \( 1 - 3.65T + 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
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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
−7.897695479893587416200040533843, −7.30275726561917238259364508417, −6.45538362131643371947183161483, −5.95036453995006520391470918803, −5.27056596531922236513435770564, −4.50020216978765695543838520385, −4.13459012363723000182651564239, −2.44746532013663791770704567248, −1.80316539406141680868675941067, −0.57638194209339431069289446824,
0.57638194209339431069289446824, 1.80316539406141680868675941067, 2.44746532013663791770704567248, 4.13459012363723000182651564239, 4.50020216978765695543838520385, 5.27056596531922236513435770564, 5.95036453995006520391470918803, 6.45538362131643371947183161483, 7.30275726561917238259364508417, 7.897695479893587416200040533843