| L(s) = 1 | − 8.01·5-s − 0.937·7-s − 5.33·11-s − 3.11·13-s + 132.·17-s + 86.6·19-s − 41.2·23-s − 60.7·25-s − 202.·29-s − 318.·31-s + 7.51·35-s − 363.·37-s + 9.83·41-s + 362.·43-s + 75.1·47-s − 342.·49-s + 403.·53-s + 42.7·55-s + 430.·59-s + 321.·61-s + 24.9·65-s + 726.·67-s + 829.·71-s − 160.·73-s + 5.00·77-s + 924.·79-s − 511.·83-s + ⋯ |
| L(s) = 1 | − 0.716·5-s − 0.0506·7-s − 0.146·11-s − 0.0664·13-s + 1.89·17-s + 1.04·19-s − 0.373·23-s − 0.485·25-s − 1.29·29-s − 1.84·31-s + 0.0362·35-s − 1.61·37-s + 0.0374·41-s + 1.28·43-s + 0.233·47-s − 0.997·49-s + 1.04·53-s + 0.104·55-s + 0.950·59-s + 0.674·61-s + 0.0476·65-s + 1.32·67-s + 1.38·71-s − 0.256·73-s + 0.00740·77-s + 1.31·79-s − 0.676·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.589562952\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.589562952\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 8.01T + 125T^{2} \) |
| 7 | \( 1 + 0.937T + 343T^{2} \) |
| 11 | \( 1 + 5.33T + 1.33e3T^{2} \) |
| 13 | \( 1 + 3.11T + 2.19e3T^{2} \) |
| 17 | \( 1 - 132.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 86.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 41.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 202.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 318.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 363.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 9.83T + 6.89e4T^{2} \) |
| 43 | \( 1 - 362.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 75.1T + 1.03e5T^{2} \) |
| 53 | \( 1 - 403.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 430.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 321.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 726.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 829.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 160.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 924.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 511.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 320.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 601.T + 9.12e5T^{2} \) |
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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
−9.395357662796211701670858546948, −8.336744713141572297764184881004, −7.57081832986982738402077617764, −7.14344709738642019578990715917, −5.68461472594867633907295988959, −5.28423016698293900623484182814, −3.80953081231420758032251734758, −3.41214375401778915982596865787, −1.93376620439237247648524773191, −0.63493654347256338631075882669,
0.63493654347256338631075882669, 1.93376620439237247648524773191, 3.41214375401778915982596865787, 3.80953081231420758032251734758, 5.28423016698293900623484182814, 5.68461472594867633907295988959, 7.14344709738642019578990715917, 7.57081832986982738402077617764, 8.336744713141572297764184881004, 9.395357662796211701670858546948
