| L(s) = 1 | − 3-s + 1.06·5-s − 7-s + 9-s − 2.72·11-s + 1.56·13-s − 1.06·15-s + 4.83·17-s − 19-s + 21-s + 2.72·23-s − 3.85·25-s − 27-s − 1.65·29-s − 4.72·31-s + 2.72·33-s − 1.06·35-s + 10.0·37-s − 1.56·39-s − 6.33·41-s + 8.76·43-s + 1.06·45-s + 9.79·47-s + 49-s − 4.83·51-s − 10.5·53-s − 2.90·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.477·5-s − 0.377·7-s + 0.333·9-s − 0.821·11-s + 0.435·13-s − 0.275·15-s + 1.17·17-s − 0.229·19-s + 0.218·21-s + 0.567·23-s − 0.771·25-s − 0.192·27-s − 0.307·29-s − 0.848·31-s + 0.474·33-s − 0.180·35-s + 1.65·37-s − 0.251·39-s − 0.989·41-s + 1.33·43-s + 0.159·45-s + 1.42·47-s + 0.142·49-s − 0.677·51-s − 1.45·53-s − 0.392·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.518750672\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.518750672\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 1.06T + 5T^{2} \) |
| 11 | \( 1 + 2.72T + 11T^{2} \) |
| 13 | \( 1 - 1.56T + 13T^{2} \) |
| 17 | \( 1 - 4.83T + 17T^{2} \) |
| 23 | \( 1 - 2.72T + 23T^{2} \) |
| 29 | \( 1 + 1.65T + 29T^{2} \) |
| 31 | \( 1 + 4.72T + 31T^{2} \) |
| 37 | \( 1 - 10.0T + 37T^{2} \) |
| 41 | \( 1 + 6.33T + 41T^{2} \) |
| 43 | \( 1 - 8.76T + 43T^{2} \) |
| 47 | \( 1 - 9.79T + 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 + 5.44T + 59T^{2} \) |
| 61 | \( 1 + 1.77T + 61T^{2} \) |
| 67 | \( 1 + 8.92T + 67T^{2} \) |
| 71 | \( 1 - 9.26T + 71T^{2} \) |
| 73 | \( 1 - 8.76T + 73T^{2} \) |
| 79 | \( 1 + 3.01T + 79T^{2} \) |
| 83 | \( 1 - 8.97T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 + 5.83T + 97T^{2} \) |
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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
−7.78303232817190393669270412422, −7.45492159122637896407722156179, −6.42407613619272470537132263585, −5.84428560572335408083360892293, −5.39570659041926441494557086938, −4.49919041497010732753465987712, −3.60304467203770059505512798966, −2.76647288009502644132400556475, −1.76301617134691467785093461596, −0.66240686522731780146336473024,
0.66240686522731780146336473024, 1.76301617134691467785093461596, 2.76647288009502644132400556475, 3.60304467203770059505512798966, 4.49919041497010732753465987712, 5.39570659041926441494557086938, 5.84428560572335408083360892293, 6.42407613619272470537132263585, 7.45492159122637896407722156179, 7.78303232817190393669270412422
