| L(s) = 1 | + 2.40·3-s + 1.57·5-s + 2.69·7-s + 2.78·9-s − 1.76·11-s + 5.06·13-s + 3.78·15-s + 0.898·17-s + 2.43·19-s + 6.48·21-s − 0.226·23-s − 2.52·25-s − 0.506·27-s − 6.55·29-s + 1.68·31-s − 4.24·33-s + 4.24·35-s + 3.49·37-s + 12.1·39-s + 5.18·41-s + 3.75·43-s + 4.39·45-s + 10.7·47-s + 0.270·49-s + 2.16·51-s + 2.30·53-s − 2.77·55-s + ⋯ |
| L(s) = 1 | + 1.38·3-s + 0.703·5-s + 1.01·7-s + 0.929·9-s − 0.532·11-s + 1.40·13-s + 0.977·15-s + 0.217·17-s + 0.559·19-s + 1.41·21-s − 0.0472·23-s − 0.504·25-s − 0.0973·27-s − 1.21·29-s + 0.302·31-s − 0.739·33-s + 0.717·35-s + 0.574·37-s + 1.95·39-s + 0.809·41-s + 0.572·43-s + 0.654·45-s + 1.57·47-s + 0.0386·49-s + 0.302·51-s + 0.316·53-s − 0.374·55-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
| \(L(1)\) |
\(\approx\) |
\(4.371448129\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.371448129\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.40T + 3T^{2} \) |
| 5 | \( 1 - 1.57T + 5T^{2} \) |
| 7 | \( 1 - 2.69T + 7T^{2} \) |
| 11 | \( 1 + 1.76T + 11T^{2} \) |
| 13 | \( 1 - 5.06T + 13T^{2} \) |
| 17 | \( 1 - 0.898T + 17T^{2} \) |
| 19 | \( 1 - 2.43T + 19T^{2} \) |
| 23 | \( 1 + 0.226T + 23T^{2} \) |
| 29 | \( 1 + 6.55T + 29T^{2} \) |
| 31 | \( 1 - 1.68T + 31T^{2} \) |
| 37 | \( 1 - 3.49T + 37T^{2} \) |
| 41 | \( 1 - 5.18T + 41T^{2} \) |
| 43 | \( 1 - 3.75T + 43T^{2} \) |
| 47 | \( 1 - 10.7T + 47T^{2} \) |
| 53 | \( 1 - 2.30T + 53T^{2} \) |
| 59 | \( 1 + 3.33T + 59T^{2} \) |
| 61 | \( 1 + 3.04T + 61T^{2} \) |
| 67 | \( 1 + 12.5T + 67T^{2} \) |
| 71 | \( 1 - 4.75T + 71T^{2} \) |
| 73 | \( 1 + 15.8T + 73T^{2} \) |
| 79 | \( 1 + 3.22T + 79T^{2} \) |
| 83 | \( 1 + 4.87T + 83T^{2} \) |
| 89 | \( 1 - 6.89T + 89T^{2} \) |
| 97 | \( 1 + 5.69T + 97T^{2} \) |
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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
−8.548549689403547049482920639951, −7.67933164273657975173455793993, −7.47064453769392975172990838627, −6.01025492209889387720856549809, −5.64179677198351481321298371687, −4.49293338756556933242483757564, −3.73711099725840558352944143152, −2.85343429816570043865772256146, −2.03731252490516140889208297612, −1.27387172171717235588174257326,
1.27387172171717235588174257326, 2.03731252490516140889208297612, 2.85343429816570043865772256146, 3.73711099725840558352944143152, 4.49293338756556933242483757564, 5.64179677198351481321298371687, 6.01025492209889387720856549809, 7.47064453769392975172990838627, 7.67933164273657975173455793993, 8.548549689403547049482920639951
