L(s) = 1 | + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s − 13-s − 14-s + 16-s + 17-s − 19-s − 20-s − 23-s + 25-s − 26-s − 28-s + 29-s + 31-s + 32-s + 34-s + 35-s + 37-s − 38-s − 40-s + 41-s − 43-s − 46-s + ⋯ |
L(s) = 1 | + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s − 13-s − 14-s + 16-s + 17-s − 19-s − 20-s − 23-s + 25-s − 26-s − 28-s + 29-s + 31-s + 32-s + 34-s + 35-s + 37-s − 38-s − 40-s + 41-s − 43-s − 46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.092291503\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.092291503\) |
\(L(1)\) |
\(\approx\) |
\(1.332797188\) |
\(L(1)\) |
\(\approx\) |
\(1.332797188\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−36.22670197160228655384957164617, −34.86979804370352953047987188318, −34.00052669826299603894318662317, −32.21508652100848086108238717144, −31.862053132219145424055672895195, −30.40972530504604648822877421808, −29.355507119523677931781543565610, −27.97065295685982701358440997141, −26.33665201440011595428737732531, −25.01194416737648581081374969192, −23.6330678372623130386463316360, −22.8032303504500431023975484460, −21.57562960541965370264093450500, −19.95148065570969446067032285389, −19.16857991604524510390831899173, −16.71025798128473101799872177954, −15.66217617484298026176809238344, −14.446117889281578588679046529210, −12.76111504593316981580874395970, −11.870654092261445677538594314170, −10.16870309638781907932950559899, −7.80326429364573470762626937960, −6.36152504478879300170016681928, −4.45916450759625444117471487542, −2.99695093746347198358466896022,
2.99695093746347198358466896022, 4.45916450759625444117471487542, 6.36152504478879300170016681928, 7.80326429364573470762626937960, 10.16870309638781907932950559899, 11.870654092261445677538594314170, 12.76111504593316981580874395970, 14.446117889281578588679046529210, 15.66217617484298026176809238344, 16.71025798128473101799872177954, 19.16857991604524510390831899173, 19.95148065570969446067032285389, 21.57562960541965370264093450500, 22.8032303504500431023975484460, 23.6330678372623130386463316360, 25.01194416737648581081374969192, 26.33665201440011595428737732531, 27.97065295685982701358440997141, 29.355507119523677931781543565610, 30.40972530504604648822877421808, 31.862053132219145424055672895195, 32.21508652100848086108238717144, 34.00052669826299603894318662317, 34.86979804370352953047987188318, 36.22670197160228655384957164617