| L(s) = 1 | − 2-s − 1.36·3-s + 4-s + 1.36·6-s − 0.636·7-s − 8-s − 1.14·9-s + 3.50·11-s − 1.36·12-s + 0.141·13-s + 0.636·14-s + 16-s − 2.14·17-s + 1.14·18-s + 19-s + 0.867·21-s − 3.50·22-s − 4.91·23-s + 1.36·24-s − 0.141·26-s + 5.64·27-s − 0.636·28-s + 7.15·29-s − 7.78·31-s − 32-s − 4.77·33-s + 2.14·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.787·3-s + 0.5·4-s + 0.556·6-s − 0.240·7-s − 0.353·8-s − 0.380·9-s + 1.05·11-s − 0.393·12-s + 0.0391·13-s + 0.170·14-s + 0.250·16-s − 0.519·17-s + 0.269·18-s + 0.229·19-s + 0.189·21-s − 0.747·22-s − 1.02·23-s + 0.278·24-s − 0.0277·26-s + 1.08·27-s − 0.120·28-s + 1.32·29-s − 1.39·31-s − 0.176·32-s − 0.831·33-s + 0.367·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 1.36T + 3T^{2} \) |
| 7 | \( 1 + 0.636T + 7T^{2} \) |
| 11 | \( 1 - 3.50T + 11T^{2} \) |
| 13 | \( 1 - 0.141T + 13T^{2} \) |
| 17 | \( 1 + 2.14T + 17T^{2} \) |
| 23 | \( 1 + 4.91T + 23T^{2} \) |
| 29 | \( 1 - 7.15T + 29T^{2} \) |
| 31 | \( 1 + 7.78T + 31T^{2} \) |
| 37 | \( 1 + 3.27T + 37T^{2} \) |
| 41 | \( 1 + 4.23T + 41T^{2} \) |
| 43 | \( 1 + 2.49T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 + 8.14T + 53T^{2} \) |
| 59 | \( 1 + 5.64T + 59T^{2} \) |
| 61 | \( 1 + 6.49T + 61T^{2} \) |
| 67 | \( 1 + 8.37T + 67T^{2} \) |
| 71 | \( 1 + 8.95T + 71T^{2} \) |
| 73 | \( 1 - 3.69T + 73T^{2} \) |
| 79 | \( 1 + 4.17T + 79T^{2} \) |
| 83 | \( 1 - 9.00T + 83T^{2} \) |
| 89 | \( 1 + 6.77T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
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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
−9.570129306125444829423161640153, −8.893607415941541781459788532898, −8.060076202098181060261049926615, −6.91417056560953214042847626485, −6.33009541371317790535834993373, −5.49942535587784043290781602513, −4.28342306821144394432530696198, −3.04008517123061672221722669610, −1.55564356243632533712808947973, 0,
1.55564356243632533712808947973, 3.04008517123061672221722669610, 4.28342306821144394432530696198, 5.49942535587784043290781602513, 6.33009541371317790535834993373, 6.91417056560953214042847626485, 8.060076202098181060261049926615, 8.893607415941541781459788532898, 9.570129306125444829423161640153
