L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 8.38·5-s − 6·6-s + 16.0·7-s + 8·8-s + 9·9-s − 16.7·10-s − 27.9·11-s − 12·12-s + 32.6·13-s + 32.0·14-s + 25.1·15-s + 16·16-s − 19.2·17-s + 18·18-s − 33.5·20-s − 48.1·21-s − 55.9·22-s + 79.7·23-s − 24·24-s − 54.6·25-s + 65.2·26-s − 27·27-s + 64.1·28-s − 141.·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.750·5-s − 0.408·6-s + 0.866·7-s + 0.353·8-s + 0.333·9-s − 0.530·10-s − 0.766·11-s − 0.288·12-s + 0.695·13-s + 0.612·14-s + 0.433·15-s + 0.250·16-s − 0.274·17-s + 0.235·18-s − 0.375·20-s − 0.500·21-s − 0.542·22-s + 0.722·23-s − 0.204·24-s − 0.437·25-s + 0.491·26-s − 0.192·27-s + 0.433·28-s − 0.905·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 + 3T \) |
| 19 | \( 1 \) |
good | 5 | \( 1 + 8.38T + 125T^{2} \) |
| 7 | \( 1 - 16.0T + 343T^{2} \) |
| 11 | \( 1 + 27.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 32.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 19.2T + 4.91e3T^{2} \) |
| 23 | \( 1 - 79.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 141.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 167.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 55.2T + 5.06e4T^{2} \) |
| 41 | \( 1 + 42.0T + 6.89e4T^{2} \) |
| 43 | \( 1 - 110.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 268.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 111.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 644.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 360.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 94.7T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.18e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 817.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 970.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.13e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 93.7T + 7.04e5T^{2} \) |
| 97 | \( 1 - 554.T + 9.12e5T^{2} \) |
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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
−8.128459163871517727517554370159, −7.46776099227250386877189362632, −6.78398113849472023922489878942, −5.64130327980054008510890579091, −5.23601990224900266390212701709, −4.25418457955200131399712345707, −3.65063691047844757453046638629, −2.39727951639062267013153402195, −1.29710507825787636151959463935, 0,
1.29710507825787636151959463935, 2.39727951639062267013153402195, 3.65063691047844757453046638629, 4.25418457955200131399712345707, 5.23601990224900266390212701709, 5.64130327980054008510890579091, 6.78398113849472023922489878942, 7.46776099227250386877189362632, 8.128459163871517727517554370159