L(s) = 1 | − 2-s − 3-s + 4-s − 2.19·5-s + 6-s − 0.720·7-s − 8-s + 9-s + 2.19·10-s + 11-s − 12-s + 5.69·13-s + 0.720·14-s + 2.19·15-s + 16-s + 4.45·17-s − 18-s − 5.59·19-s − 2.19·20-s + 0.720·21-s − 22-s + 9.41·23-s + 24-s − 0.178·25-s − 5.69·26-s − 27-s − 0.720·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.981·5-s + 0.408·6-s − 0.272·7-s − 0.353·8-s + 0.333·9-s + 0.694·10-s + 0.301·11-s − 0.288·12-s + 1.58·13-s + 0.192·14-s + 0.566·15-s + 0.250·16-s + 1.08·17-s − 0.235·18-s − 1.28·19-s − 0.490·20-s + 0.157·21-s − 0.213·22-s + 1.96·23-s + 0.204·24-s − 0.0357·25-s − 1.11·26-s − 0.192·27-s − 0.136·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8804165584\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8804165584\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
good | 5 | \( 1 + 2.19T + 5T^{2} \) |
| 7 | \( 1 + 0.720T + 7T^{2} \) |
| 13 | \( 1 - 5.69T + 13T^{2} \) |
| 17 | \( 1 - 4.45T + 17T^{2} \) |
| 19 | \( 1 + 5.59T + 19T^{2} \) |
| 23 | \( 1 - 9.41T + 23T^{2} \) |
| 29 | \( 1 - 2.40T + 29T^{2} \) |
| 31 | \( 1 - 3.92T + 31T^{2} \) |
| 37 | \( 1 + 2.96T + 37T^{2} \) |
| 41 | \( 1 + 7.55T + 41T^{2} \) |
| 43 | \( 1 + 9.82T + 43T^{2} \) |
| 47 | \( 1 - 5.15T + 47T^{2} \) |
| 53 | \( 1 - 6.87T + 53T^{2} \) |
| 59 | \( 1 - 5.27T + 59T^{2} \) |
| 67 | \( 1 - 3.37T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 + 8.19T + 73T^{2} \) |
| 79 | \( 1 - 3.03T + 79T^{2} \) |
| 83 | \( 1 - 4.92T + 83T^{2} \) |
| 89 | \( 1 + 6.58T + 89T^{2} \) |
| 97 | \( 1 + 11.0T + 97T^{2} \) |
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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
−8.589255830515479836730353372492, −7.79811465313315581250733586747, −6.91205074009015274135718345903, −6.48398055718498678277411477138, −5.62668188304738720876722400060, −4.65375497934086276962602564931, −3.70941387882118597546688903274, −3.11108974455509434064019981662, −1.56035326036788537971335949864, −0.65482578521508102995810035197,
0.65482578521508102995810035197, 1.56035326036788537971335949864, 3.11108974455509434064019981662, 3.70941387882118597546688903274, 4.65375497934086276962602564931, 5.62668188304738720876722400060, 6.48398055718498678277411477138, 6.91205074009015274135718345903, 7.79811465313315581250733586747, 8.589255830515479836730353372492