L(s) = 1 | + 2-s − 3-s + 4-s − 3.41·5-s − 6-s + 8-s + 9-s − 3.41·10-s + 2.41·11-s − 12-s + 13-s + 3.41·15-s + 16-s + 0.414·17-s + 18-s − 7.82·19-s − 3.41·20-s + 2.41·22-s − 1.41·23-s − 24-s + 6.65·25-s + 26-s − 27-s + 3.82·29-s + 3.41·30-s + 8.48·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.52·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 1.07·10-s + 0.727·11-s − 0.288·12-s + 0.277·13-s + 0.881·15-s + 0.250·16-s + 0.100·17-s + 0.235·18-s − 1.79·19-s − 0.763·20-s + 0.514·22-s − 0.294·23-s − 0.204·24-s + 1.33·25-s + 0.196·26-s − 0.192·27-s + 0.710·29-s + 0.623·30-s + 1.52·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 5 | \( 1 + 3.41T + 5T^{2} \) |
| 11 | \( 1 - 2.41T + 11T^{2} \) |
| 17 | \( 1 - 0.414T + 17T^{2} \) |
| 19 | \( 1 + 7.82T + 19T^{2} \) |
| 23 | \( 1 + 1.41T + 23T^{2} \) |
| 29 | \( 1 - 3.82T + 29T^{2} \) |
| 31 | \( 1 - 8.48T + 31T^{2} \) |
| 37 | \( 1 + 1.41T + 37T^{2} \) |
| 41 | \( 1 - 9.89T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 - T + 47T^{2} \) |
| 53 | \( 1 + 7.48T + 53T^{2} \) |
| 59 | \( 1 + 12.0T + 59T^{2} \) |
| 61 | \( 1 + 1.58T + 61T^{2} \) |
| 67 | \( 1 + 3.82T + 67T^{2} \) |
| 71 | \( 1 + 5T + 71T^{2} \) |
| 73 | \( 1 + 1.41T + 73T^{2} \) |
| 79 | \( 1 + 0.343T + 79T^{2} \) |
| 83 | \( 1 + 3.65T + 83T^{2} \) |
| 89 | \( 1 - 5.41T + 89T^{2} \) |
| 97 | \( 1 + 15.0T + 97T^{2} \) |
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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
−8.113413204190550388226751430637, −7.24778851651579104131222721478, −6.48128066040753662524037487260, −6.07918039980903534585299138554, −4.76772038847933459310054871518, −4.35810767562569608239089280948, −3.72826531637788830471425767191, −2.75497545887510073395748543822, −1.35541431671129224739576996298, 0,
1.35541431671129224739576996298, 2.75497545887510073395748543822, 3.72826531637788830471425767191, 4.35810767562569608239089280948, 4.76772038847933459310054871518, 6.07918039980903534585299138554, 6.48128066040753662524037487260, 7.24778851651579104131222721478, 8.113413204190550388226751430637