| L(s) = 1 | + 3·3-s − 10.3·5-s + 18.4·7-s + 9·9-s + 93.4·13-s − 30.9·15-s − 75.0·17-s + 38.2·19-s + 55.2·21-s − 107.·23-s − 18.8·25-s + 27·27-s + 185.·29-s + 285.·31-s − 189.·35-s + 35.8·37-s + 280.·39-s − 446.·41-s + 295.·43-s − 92.7·45-s − 66.3·47-s − 4.12·49-s − 225.·51-s + 430.·53-s + 114.·57-s − 382.·59-s + 277.·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.921·5-s + 0.993·7-s + 0.333·9-s + 1.99·13-s − 0.532·15-s − 1.07·17-s + 0.461·19-s + 0.573·21-s − 0.973·23-s − 0.150·25-s + 0.192·27-s + 1.18·29-s + 1.65·31-s − 0.916·35-s + 0.159·37-s + 1.15·39-s − 1.69·41-s + 1.04·43-s − 0.307·45-s − 0.206·47-s − 0.0120·49-s − 0.618·51-s + 1.11·53-s + 0.266·57-s − 0.844·59-s + 0.582·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.834354134\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.834354134\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 10.3T + 125T^{2} \) |
| 7 | \( 1 - 18.4T + 343T^{2} \) |
| 13 | \( 1 - 93.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 75.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 38.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 107.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 185.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 285.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 35.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 446.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 295.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 66.3T + 1.03e5T^{2} \) |
| 53 | \( 1 - 430.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 382.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 277.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 363.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 991.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 223.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 310.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.07e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 519.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.09e3T + 9.12e5T^{2} \) |
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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
−8.719618791303312960077674403684, −8.401274162404501390068298805364, −7.84382692680662178305256979658, −6.79684170948227500311499916023, −5.95702686102702255678771948414, −4.64469852346036868989481359335, −4.07887964698170825706428560282, −3.17482008660312470183075688649, −1.89796751765664286378818172580, −0.843433896825307915882105291124,
0.843433896825307915882105291124, 1.89796751765664286378818172580, 3.17482008660312470183075688649, 4.07887964698170825706428560282, 4.64469852346036868989481359335, 5.95702686102702255678771948414, 6.79684170948227500311499916023, 7.84382692680662178305256979658, 8.401274162404501390068298805364, 8.719618791303312960077674403684
