| L(s) = 1 | − 0.414·3-s − 5-s + 0.585·7-s − 2.82·9-s + 11-s − 6.24·13-s + 0.414·15-s + 0.585·17-s + 19-s − 0.242·21-s + 3·23-s − 4·25-s + 2.41·27-s + 2.24·29-s + 3.58·31-s − 0.414·33-s − 0.585·35-s − 4.07·37-s + 2.58·39-s + 9.65·41-s − 11.6·43-s + 2.82·45-s − 3.17·47-s − 6.65·49-s − 0.242·51-s + 12.4·53-s − 55-s + ⋯ |
| L(s) = 1 | − 0.239·3-s − 0.447·5-s + 0.221·7-s − 0.942·9-s + 0.301·11-s − 1.73·13-s + 0.106·15-s + 0.142·17-s + 0.229·19-s − 0.0529·21-s + 0.625·23-s − 0.800·25-s + 0.464·27-s + 0.416·29-s + 0.644·31-s − 0.0721·33-s − 0.0990·35-s − 0.669·37-s + 0.414·39-s + 1.50·41-s − 1.77·43-s + 0.421·45-s − 0.462·47-s − 0.950·49-s − 0.0339·51-s + 1.71·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.080467492\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.080467492\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 0.414T + 3T^{2} \) |
| 5 | \( 1 + T + 5T^{2} \) |
| 7 | \( 1 - 0.585T + 7T^{2} \) |
| 13 | \( 1 + 6.24T + 13T^{2} \) |
| 17 | \( 1 - 0.585T + 17T^{2} \) |
| 23 | \( 1 - 3T + 23T^{2} \) |
| 29 | \( 1 - 2.24T + 29T^{2} \) |
| 31 | \( 1 - 3.58T + 31T^{2} \) |
| 37 | \( 1 + 4.07T + 37T^{2} \) |
| 41 | \( 1 - 9.65T + 41T^{2} \) |
| 43 | \( 1 + 11.6T + 43T^{2} \) |
| 47 | \( 1 + 3.17T + 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 - 4.41T + 59T^{2} \) |
| 61 | \( 1 - 3.07T + 61T^{2} \) |
| 67 | \( 1 - 7.58T + 67T^{2} \) |
| 71 | \( 1 - 9.58T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 - 17.4T + 79T^{2} \) |
| 83 | \( 1 + 0.585T + 83T^{2} \) |
| 89 | \( 1 + 14.8T + 89T^{2} \) |
| 97 | \( 1 + 0.414T + 97T^{2} \) |
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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
−8.426409135054354923970304159696, −7.991997465755921226993863108002, −7.09844474638786329005978224641, −6.49462117806547018368736638153, −5.36513540480948750595088303906, −4.99219806794224664260243035308, −3.96214201875540012979904580391, −2.99300249444462046182484527231, −2.14941078430671813112078324147, −0.60295940610007321725013167706,
0.60295940610007321725013167706, 2.14941078430671813112078324147, 2.99300249444462046182484527231, 3.96214201875540012979904580391, 4.99219806794224664260243035308, 5.36513540480948750595088303906, 6.49462117806547018368736638153, 7.09844474638786329005978224641, 7.991997465755921226993863108002, 8.426409135054354923970304159696
