| L(s) = 1 | − 8·2-s + 27·3-s + 64·4-s − 216·6-s − 512·7-s − 512·8-s + 729·9-s + 5.46e3·11-s + 1.72e3·12-s − 1.01e4·13-s + 4.09e3·14-s + 4.09e3·16-s + 9.91e3·17-s − 5.83e3·18-s − 1.24e4·19-s − 1.38e4·21-s − 4.36e4·22-s − 3.36e4·23-s − 1.38e4·24-s + 8.13e4·26-s + 1.96e4·27-s − 3.27e4·28-s − 1.87e5·29-s − 4.25e4·31-s − 3.27e4·32-s + 1.47e5·33-s − 7.93e4·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.564·7-s − 0.353·8-s + 1/3·9-s + 1.23·11-s + 0.288·12-s − 1.28·13-s + 0.398·14-s + 1/4·16-s + 0.489·17-s − 0.235·18-s − 0.415·19-s − 0.325·21-s − 0.874·22-s − 0.575·23-s − 0.204·24-s + 0.907·26-s + 0.192·27-s − 0.282·28-s − 1.43·29-s − 0.256·31-s − 0.176·32-s + 0.714·33-s − 0.346·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{3} T \) |
| 3 | \( 1 - p^{3} T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 512 T + p^{7} T^{2} \) |
| 11 | \( 1 - 5460 T + p^{7} T^{2} \) |
| 13 | \( 1 + 782 p T + p^{7} T^{2} \) |
| 17 | \( 1 - 9918 T + p^{7} T^{2} \) |
| 19 | \( 1 + 12436 T + p^{7} T^{2} \) |
| 23 | \( 1 + 33600 T + p^{7} T^{2} \) |
| 29 | \( 1 + 187914 T + p^{7} T^{2} \) |
| 31 | \( 1 + 42592 T + p^{7} T^{2} \) |
| 37 | \( 1 - 544066 T + p^{7} T^{2} \) |
| 41 | \( 1 - 374394 T + p^{7} T^{2} \) |
| 43 | \( 1 - 540532 T + p^{7} T^{2} \) |
| 47 | \( 1 + 1338360 T + p^{7} T^{2} \) |
| 53 | \( 1 + 1308222 T + p^{7} T^{2} \) |
| 59 | \( 1 - 262740 T + p^{7} T^{2} \) |
| 61 | \( 1 + 976330 T + p^{7} T^{2} \) |
| 67 | \( 1 + 3559172 T + p^{7} T^{2} \) |
| 71 | \( 1 + 2673720 T + p^{7} T^{2} \) |
| 73 | \( 1 - 3032134 T + p^{7} T^{2} \) |
| 79 | \( 1 + 5475808 T + p^{7} T^{2} \) |
| 83 | \( 1 + 2231556 T + p^{7} T^{2} \) |
| 89 | \( 1 + 10050678 T + p^{7} T^{2} \) |
| 97 | \( 1 + 5727554 T + p^{7} T^{2} \) |
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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
−11.08287359949303529558964975263, −9.600714192342410731154112006057, −9.484733101564917938244543688986, −8.060411917000722438173049793890, −7.14211651486588304452907591153, −6.02591763904451784966786363626, −4.21531149264157716128801045727, −2.88683297471762692762184747445, −1.57948950283318796012569317568, 0,
1.57948950283318796012569317568, 2.88683297471762692762184747445, 4.21531149264157716128801045727, 6.02591763904451784966786363626, 7.14211651486588304452907591153, 8.060411917000722438173049793890, 9.484733101564917938244543688986, 9.600714192342410731154112006057, 11.08287359949303529558964975263
