L(s) = 1 | + 1.53·3-s + 5.03·7-s − 0.633·9-s + 3.03·11-s + 4.57·13-s + 1.07·17-s − 19-s + 7.74·21-s + 4.11·23-s − 5.58·27-s − 1.07·29-s − 5.58·31-s + 4.66·33-s − 0.0947·37-s + 7.03·39-s + 10.6·41-s + 5.03·43-s − 12.2·47-s + 18.3·49-s + 1.65·51-s + 4.09·53-s − 1.53·57-s + 1.39·59-s − 5.69·61-s − 3.18·63-s + 5.28·67-s + 6.32·69-s + ⋯ |
L(s) = 1 | + 0.888·3-s + 1.90·7-s − 0.211·9-s + 0.914·11-s + 1.26·13-s + 0.261·17-s − 0.229·19-s + 1.68·21-s + 0.857·23-s − 1.07·27-s − 0.199·29-s − 1.00·31-s + 0.812·33-s − 0.0155·37-s + 1.12·39-s + 1.66·41-s + 0.767·43-s − 1.79·47-s + 2.61·49-s + 0.231·51-s + 0.562·53-s − 0.203·57-s + 0.182·59-s − 0.729·61-s − 0.401·63-s + 0.645·67-s + 0.761·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.235956293\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.235956293\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 - 1.53T + 3T^{2} \) |
| 7 | \( 1 - 5.03T + 7T^{2} \) |
| 11 | \( 1 - 3.03T + 11T^{2} \) |
| 13 | \( 1 - 4.57T + 13T^{2} \) |
| 17 | \( 1 - 1.07T + 17T^{2} \) |
| 23 | \( 1 - 4.11T + 23T^{2} \) |
| 29 | \( 1 + 1.07T + 29T^{2} \) |
| 31 | \( 1 + 5.58T + 31T^{2} \) |
| 37 | \( 1 + 0.0947T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 - 5.03T + 43T^{2} \) |
| 47 | \( 1 + 12.2T + 47T^{2} \) |
| 53 | \( 1 - 4.09T + 53T^{2} \) |
| 59 | \( 1 - 1.39T + 59T^{2} \) |
| 61 | \( 1 + 5.69T + 61T^{2} \) |
| 67 | \( 1 - 5.28T + 67T^{2} \) |
| 71 | \( 1 - 5.67T + 71T^{2} \) |
| 73 | \( 1 + 9.07T + 73T^{2} \) |
| 79 | \( 1 - 5.39T + 79T^{2} \) |
| 83 | \( 1 - 1.95T + 83T^{2} \) |
| 89 | \( 1 + 2.18T + 89T^{2} \) |
| 97 | \( 1 - 2.16T + 97T^{2} \) |
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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
−7.88067950627968385321727626750, −7.53254155431020257146628319848, −6.51105257898596533211624259478, −5.71510684513258739717011804516, −5.04431333864212017147356434928, −4.12967513917981083346706199943, −3.65578815590354439471200081997, −2.62264255388332758007980102369, −1.73545798396630404152577041266, −1.11370411177691766805520463828,
1.11370411177691766805520463828, 1.73545798396630404152577041266, 2.62264255388332758007980102369, 3.65578815590354439471200081997, 4.12967513917981083346706199943, 5.04431333864212017147356434928, 5.71510684513258739717011804516, 6.51105257898596533211624259478, 7.53254155431020257146628319848, 7.88067950627968385321727626750