| L(s) = 1 | + 2.21·3-s + 3.59·5-s − 5.06·7-s + 1.90·9-s − 2.55·11-s − 4.93·13-s + 7.96·15-s + 2.68·17-s + 4.72·19-s − 11.2·21-s − 1.49·23-s + 7.93·25-s − 2.41·27-s + 2.43·29-s + 3.19·31-s − 5.67·33-s − 18.2·35-s + 2.90·37-s − 10.9·39-s − 41-s − 7.62·43-s + 6.86·45-s + 5.15·47-s + 18.6·49-s + 5.95·51-s − 11.1·53-s − 9.20·55-s + ⋯ |
| L(s) = 1 | + 1.27·3-s + 1.60·5-s − 1.91·7-s + 0.636·9-s − 0.771·11-s − 1.36·13-s + 2.05·15-s + 0.651·17-s + 1.08·19-s − 2.44·21-s − 0.311·23-s + 1.58·25-s − 0.465·27-s + 0.451·29-s + 0.573·31-s − 0.987·33-s − 3.07·35-s + 0.478·37-s − 1.75·39-s − 0.156·41-s − 1.16·43-s + 1.02·45-s + 0.751·47-s + 2.66·49-s + 0.833·51-s − 1.52·53-s − 1.24·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.643017523\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.643017523\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 - 2.21T + 3T^{2} \) |
| 5 | \( 1 - 3.59T + 5T^{2} \) |
| 7 | \( 1 + 5.06T + 7T^{2} \) |
| 11 | \( 1 + 2.55T + 11T^{2} \) |
| 13 | \( 1 + 4.93T + 13T^{2} \) |
| 17 | \( 1 - 2.68T + 17T^{2} \) |
| 19 | \( 1 - 4.72T + 19T^{2} \) |
| 23 | \( 1 + 1.49T + 23T^{2} \) |
| 29 | \( 1 - 2.43T + 29T^{2} \) |
| 31 | \( 1 - 3.19T + 31T^{2} \) |
| 37 | \( 1 - 2.90T + 37T^{2} \) |
| 43 | \( 1 + 7.62T + 43T^{2} \) |
| 47 | \( 1 - 5.15T + 47T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 - 1.49T + 59T^{2} \) |
| 61 | \( 1 - 1.06T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 + 12.6T + 71T^{2} \) |
| 73 | \( 1 - 8.28T + 73T^{2} \) |
| 79 | \( 1 + 4.28T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 - 9.36T + 89T^{2} \) |
| 97 | \( 1 + 3.37T + 97T^{2} \) |
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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
−13.15427140730918692295360380935, −12.29667576377047686997729092509, −10.03193261044176954620528220660, −9.883325739263117266406821137351, −9.113520033222656878036949681569, −7.67071754643379739839855280267, −6.48411868103710289674271580053, −5.35355840171940356639548895826, −3.17236024489173932430653019754, −2.46650905065687307909939729608,
2.46650905065687307909939729608, 3.17236024489173932430653019754, 5.35355840171940356639548895826, 6.48411868103710289674271580053, 7.67071754643379739839855280267, 9.113520033222656878036949681569, 9.883325739263117266406821137351, 10.03193261044176954620528220660, 12.29667576377047686997729092509, 13.15427140730918692295360380935
