| L(s) = 1 | − 2.27·3-s + 2.37·5-s + 1.75·7-s + 2.16·9-s + 6.02·11-s + 5.29·13-s − 5.40·15-s + 7.04·17-s − 6.15·19-s − 3.99·21-s + 5.05·23-s + 0.659·25-s + 1.89·27-s + 2.49·29-s + 4.57·31-s − 13.6·33-s + 4.17·35-s + 8.89·37-s − 12.0·39-s + 10.6·41-s + 11.1·43-s + 5.15·45-s − 10.2·47-s − 3.91·49-s − 16.0·51-s − 2.73·53-s + 14.3·55-s + ⋯ |
| L(s) = 1 | − 1.31·3-s + 1.06·5-s + 0.663·7-s + 0.721·9-s + 1.81·11-s + 1.46·13-s − 1.39·15-s + 1.70·17-s − 1.41·19-s − 0.870·21-s + 1.05·23-s + 0.131·25-s + 0.365·27-s + 0.463·29-s + 0.822·31-s − 2.38·33-s + 0.706·35-s + 1.46·37-s − 1.92·39-s + 1.65·41-s + 1.70·43-s + 0.767·45-s − 1.49·47-s − 0.559·49-s − 2.24·51-s − 0.375·53-s + 1.93·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.220421137\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.220421137\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 + 2.27T + 3T^{2} \) |
| 5 | \( 1 - 2.37T + 5T^{2} \) |
| 7 | \( 1 - 1.75T + 7T^{2} \) |
| 11 | \( 1 - 6.02T + 11T^{2} \) |
| 13 | \( 1 - 5.29T + 13T^{2} \) |
| 17 | \( 1 - 7.04T + 17T^{2} \) |
| 19 | \( 1 + 6.15T + 19T^{2} \) |
| 23 | \( 1 - 5.05T + 23T^{2} \) |
| 29 | \( 1 - 2.49T + 29T^{2} \) |
| 31 | \( 1 - 4.57T + 31T^{2} \) |
| 37 | \( 1 - 8.89T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 + 2.73T + 53T^{2} \) |
| 59 | \( 1 + 13.9T + 59T^{2} \) |
| 61 | \( 1 + 8.80T + 61T^{2} \) |
| 67 | \( 1 + 9.78T + 67T^{2} \) |
| 71 | \( 1 + 8.39T + 71T^{2} \) |
| 73 | \( 1 - 7.86T + 73T^{2} \) |
| 79 | \( 1 + 2.93T + 79T^{2} \) |
| 83 | \( 1 + 9.64T + 83T^{2} \) |
| 89 | \( 1 + 18.6T + 89T^{2} \) |
| 97 | \( 1 - 6.29T + 97T^{2} \) |
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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
−8.545335145345342503415545165846, −7.65081812017354742607684775147, −6.52059329199522736031092706572, −6.04545483967017534295957765180, −5.89110607971903527673881740483, −4.71174037293208988283713618434, −4.15296230001661483809494910942, −2.93579592394020269051156807574, −1.37052699466579901088695218110, −1.17366615504668058429529773485,
1.17366615504668058429529773485, 1.37052699466579901088695218110, 2.93579592394020269051156807574, 4.15296230001661483809494910942, 4.71174037293208988283713618434, 5.89110607971903527673881740483, 6.04545483967017534295957765180, 6.52059329199522736031092706572, 7.65081812017354742607684775147, 8.545335145345342503415545165846
