| L(s) = 1 | + 2-s + 4-s + 5-s + 4.56·7-s + 8-s + 10-s − 3.41·11-s + 0.971·13-s + 4.56·14-s + 16-s − 2.56·17-s + 4.86·19-s + 20-s − 3.41·22-s + 1.02·23-s + 25-s + 0.971·26-s + 4.56·28-s − 3.59·29-s + 2.55·31-s + 32-s − 2.56·34-s + 4.56·35-s + 37-s + 4.86·38-s + 40-s + 7.74·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.447·5-s + 1.72·7-s + 0.353·8-s + 0.316·10-s − 1.03·11-s + 0.269·13-s + 1.22·14-s + 0.250·16-s − 0.622·17-s + 1.11·19-s + 0.223·20-s − 0.728·22-s + 0.214·23-s + 0.200·25-s + 0.190·26-s + 0.863·28-s − 0.667·29-s + 0.458·31-s + 0.176·32-s − 0.440·34-s + 0.772·35-s + 0.164·37-s + 0.789·38-s + 0.158·40-s + 1.20·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.995415292\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.995415292\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| good | 7 | \( 1 - 4.56T + 7T^{2} \) |
| 11 | \( 1 + 3.41T + 11T^{2} \) |
| 13 | \( 1 - 0.971T + 13T^{2} \) |
| 17 | \( 1 + 2.56T + 17T^{2} \) |
| 19 | \( 1 - 4.86T + 19T^{2} \) |
| 23 | \( 1 - 1.02T + 23T^{2} \) |
| 29 | \( 1 + 3.59T + 29T^{2} \) |
| 31 | \( 1 - 2.55T + 31T^{2} \) |
| 41 | \( 1 - 7.74T + 41T^{2} \) |
| 43 | \( 1 - 9.74T + 43T^{2} \) |
| 47 | \( 1 - 4.99T + 47T^{2} \) |
| 53 | \( 1 + 6.71T + 53T^{2} \) |
| 59 | \( 1 - 6.83T + 59T^{2} \) |
| 61 | \( 1 + 6.99T + 61T^{2} \) |
| 67 | \( 1 + 15.9T + 67T^{2} \) |
| 71 | \( 1 + 8.29T + 71T^{2} \) |
| 73 | \( 1 - 1.56T + 73T^{2} \) |
| 79 | \( 1 + 6.29T + 79T^{2} \) |
| 83 | \( 1 - 7.86T + 83T^{2} \) |
| 89 | \( 1 + 17.3T + 89T^{2} \) |
| 97 | \( 1 + 0.747T + 97T^{2} \) |
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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
−8.544025536703326258670891720799, −7.57513071058190581004678604830, −7.43470489998096180473278523431, −6.05511715322896558440639280924, −5.52294127835059554379639219806, −4.80061832045932372725559662541, −4.22515266862683460625669271459, −2.93353842991766621689934137868, −2.15604879615561611124697767487, −1.17974416425608377846294685597,
1.17974416425608377846294685597, 2.15604879615561611124697767487, 2.93353842991766621689934137868, 4.22515266862683460625669271459, 4.80061832045932372725559662541, 5.52294127835059554379639219806, 6.05511715322896558440639280924, 7.43470489998096180473278523431, 7.57513071058190581004678604830, 8.544025536703326258670891720799
