| L(s) = 1 | + 2-s + 3.38·3-s + 4-s − 5-s + 3.38·6-s + 8-s + 8.42·9-s − 10-s − 4.17·11-s + 3.38·12-s + 3.15·13-s − 3.38·15-s + 16-s − 17-s + 8.42·18-s + 7.10·19-s − 20-s − 4.17·22-s − 3.81·23-s + 3.38·24-s + 25-s + 3.15·26-s + 18.3·27-s + 0.468·29-s − 3.38·30-s − 1.30·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.95·3-s + 0.5·4-s − 0.447·5-s + 1.37·6-s + 0.353·8-s + 2.80·9-s − 0.316·10-s − 1.25·11-s + 0.975·12-s + 0.874·13-s − 0.872·15-s + 0.250·16-s − 0.242·17-s + 1.98·18-s + 1.63·19-s − 0.223·20-s − 0.889·22-s − 0.795·23-s + 0.689·24-s + 0.200·25-s + 0.618·26-s + 3.52·27-s + 0.0870·29-s − 0.617·30-s − 0.233·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.028465769\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.028465769\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 3.38T + 3T^{2} \) |
| 11 | \( 1 + 4.17T + 11T^{2} \) |
| 13 | \( 1 - 3.15T + 13T^{2} \) |
| 19 | \( 1 - 7.10T + 19T^{2} \) |
| 23 | \( 1 + 3.81T + 23T^{2} \) |
| 29 | \( 1 - 0.468T + 29T^{2} \) |
| 31 | \( 1 + 1.30T + 31T^{2} \) |
| 37 | \( 1 - 4.88T + 37T^{2} \) |
| 41 | \( 1 - 9.05T + 41T^{2} \) |
| 43 | \( 1 + 6.82T + 43T^{2} \) |
| 47 | \( 1 + 11.2T + 47T^{2} \) |
| 53 | \( 1 - 7.54T + 53T^{2} \) |
| 59 | \( 1 - 1.40T + 59T^{2} \) |
| 61 | \( 1 - 15.0T + 61T^{2} \) |
| 67 | \( 1 - 2.50T + 67T^{2} \) |
| 71 | \( 1 - 3.87T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 + 6.56T + 83T^{2} \) |
| 89 | \( 1 + 5.67T + 89T^{2} \) |
| 97 | \( 1 - 4.80T + 97T^{2} \) |
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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
−7.895283596477995554840339529629, −7.33406304000428023988487050634, −6.62332683536447948621269410355, −5.56712437695153806626523766141, −4.79442389377014925568502099453, −3.98196368532665155610999105434, −3.46730487329384879705500626875, −2.81689305760845489906284429932, −2.17635806815590963347803074170, −1.13421006274759990227013616562,
1.13421006274759990227013616562, 2.17635806815590963347803074170, 2.81689305760845489906284429932, 3.46730487329384879705500626875, 3.98196368532665155610999105434, 4.79442389377014925568502099453, 5.56712437695153806626523766141, 6.62332683536447948621269410355, 7.33406304000428023988487050634, 7.895283596477995554840339529629
