L(s) = 1 | + 2-s + 0.447·3-s + 4-s − 5-s + 0.447·6-s + 1.86·7-s + 8-s − 2.79·9-s − 10-s − 4.77·11-s + 0.447·12-s − 13-s + 1.86·14-s − 0.447·15-s + 16-s + 0.778·17-s − 2.79·18-s + 1.72·19-s − 20-s + 0.834·21-s − 4.77·22-s + 7.35·23-s + 0.447·24-s + 25-s − 26-s − 2.59·27-s + 1.86·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.258·3-s + 0.5·4-s − 0.447·5-s + 0.182·6-s + 0.704·7-s + 0.353·8-s − 0.933·9-s − 0.316·10-s − 1.43·11-s + 0.129·12-s − 0.277·13-s + 0.498·14-s − 0.115·15-s + 0.250·16-s + 0.188·17-s − 0.659·18-s + 0.394·19-s − 0.223·20-s + 0.182·21-s − 1.01·22-s + 1.53·23-s + 0.0913·24-s + 0.200·25-s − 0.196·26-s − 0.499·27-s + 0.352·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.879555832\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.879555832\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 + T \) |
good | 3 | \( 1 - 0.447T + 3T^{2} \) |
| 7 | \( 1 - 1.86T + 7T^{2} \) |
| 11 | \( 1 + 4.77T + 11T^{2} \) |
| 17 | \( 1 - 0.778T + 17T^{2} \) |
| 19 | \( 1 - 1.72T + 19T^{2} \) |
| 23 | \( 1 - 7.35T + 23T^{2} \) |
| 29 | \( 1 - 4.16T + 29T^{2} \) |
| 37 | \( 1 - 3.34T + 37T^{2} \) |
| 41 | \( 1 - 9.99T + 41T^{2} \) |
| 43 | \( 1 - 3.66T + 43T^{2} \) |
| 47 | \( 1 - 13.1T + 47T^{2} \) |
| 53 | \( 1 - 4.43T + 53T^{2} \) |
| 59 | \( 1 - 0.881T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 - 12.5T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 + 3.07T + 73T^{2} \) |
| 79 | \( 1 + 6.67T + 79T^{2} \) |
| 83 | \( 1 - 9.02T + 83T^{2} \) |
| 89 | \( 1 - 1.00T + 89T^{2} \) |
| 97 | \( 1 + 0.347T + 97T^{2} \) |
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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
−8.318477914587690933476880408649, −7.61946538554596665882339629274, −7.18735764263874096947848312628, −5.98496444403688986664534046403, −5.31648814249654893504315857916, −4.81067055822541395898596609113, −3.86044413310342055539829176018, −2.79817729689008600843588397467, −2.47929094525654821875115895068, −0.854785005844326845476791014616,
0.854785005844326845476791014616, 2.47929094525654821875115895068, 2.79817729689008600843588397467, 3.86044413310342055539829176018, 4.81067055822541395898596609113, 5.31648814249654893504315857916, 5.98496444403688986664534046403, 7.18735764263874096947848312628, 7.61946538554596665882339629274, 8.318477914587690933476880408649