L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s + 20.2·5-s + 6·6-s − 22.8·7-s + 8·8-s + 9·9-s + 40.4·10-s − 57.5·11-s + 12·12-s − 27.3·13-s − 45.7·14-s + 60.6·15-s + 16·16-s − 73.4·17-s + 18·18-s + 80.8·20-s − 68.6·21-s − 115.·22-s − 188.·23-s + 24·24-s + 283.·25-s − 54.6·26-s + 27·27-s − 91.5·28-s + 7.15·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.80·5-s + 0.408·6-s − 1.23·7-s + 0.353·8-s + 0.333·9-s + 1.27·10-s − 1.57·11-s + 0.288·12-s − 0.582·13-s − 0.873·14-s + 1.04·15-s + 0.250·16-s − 1.04·17-s + 0.235·18-s + 0.903·20-s − 0.713·21-s − 1.11·22-s − 1.70·23-s + 0.204·24-s + 2.26·25-s − 0.412·26-s + 0.192·27-s − 0.617·28-s + 0.0458·29-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 | \( 1 - 20.2T + 125T^{2} \) |
| 7 | \( 1 + 22.8T + 343T^{2} \) |
| 11 | \( 1 + 57.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 27.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 73.4T + 4.91e3T^{2} \) |
| 23 | \( 1 + 188.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 7.15T + 2.43e4T^{2} \) |
| 31 | \( 1 - 117.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 332.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 42.7T + 6.89e4T^{2} \) |
| 43 | \( 1 - 47.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + 507.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 492.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 460.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 450.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 522.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 931.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 350.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 194.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 286.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 839.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 605.T + 9.12e5T^{2} \) |
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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
−8.321622349547774200657137053450, −7.44068747962895723673206560638, −6.36921030693616963690089516876, −6.12443891193494000997479251201, −5.17560852112574304966106958961, −4.35797151214545110113316873409, −2.94575367698954770313552622112, −2.59324039512907571203166969239, −1.78200284540918397062871622277, 0,
1.78200284540918397062871622277, 2.59324039512907571203166969239, 2.94575367698954770313552622112, 4.35797151214545110113316873409, 5.17560852112574304966106958961, 6.12443891193494000997479251201, 6.36921030693616963690089516876, 7.44068747962895723673206560638, 8.321622349547774200657137053450