| L(s) = 1 | + 1.61·2-s − 3-s + 0.618·4-s − 1.61·6-s − 2.23·8-s + 9-s − 2.23·11-s − 0.618·12-s + 6.47·13-s − 4.85·16-s − 2.47·17-s + 1.61·18-s − 2.47·19-s − 3.61·22-s + 4.23·23-s + 2.23·24-s + 10.4·26-s − 27-s − 3·29-s + 4·31-s − 3.38·32-s + 2.23·33-s − 4.00·34-s + 0.618·36-s − 3.47·37-s − 4.00·38-s − 6.47·39-s + ⋯ |
| L(s) = 1 | + 1.14·2-s − 0.577·3-s + 0.309·4-s − 0.660·6-s − 0.790·8-s + 0.333·9-s − 0.674·11-s − 0.178·12-s + 1.79·13-s − 1.21·16-s − 0.599·17-s + 0.381·18-s − 0.567·19-s − 0.771·22-s + 0.883·23-s + 0.456·24-s + 2.05·26-s − 0.192·27-s − 0.557·29-s + 0.718·31-s − 0.597·32-s + 0.389·33-s − 0.685·34-s + 0.103·36-s − 0.570·37-s − 0.648·38-s − 1.03·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.371935906\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.371935906\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 11 | \( 1 + 2.23T + 11T^{2} \) |
| 13 | \( 1 - 6.47T + 13T^{2} \) |
| 17 | \( 1 + 2.47T + 17T^{2} \) |
| 19 | \( 1 + 2.47T + 19T^{2} \) |
| 23 | \( 1 - 4.23T + 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 3.47T + 37T^{2} \) |
| 41 | \( 1 - 8.94T + 41T^{2} \) |
| 43 | \( 1 - 4.70T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 6.47T + 59T^{2} \) |
| 61 | \( 1 - 12T + 61T^{2} \) |
| 67 | \( 1 + 12.7T + 67T^{2} \) |
| 71 | \( 1 - 8.23T + 71T^{2} \) |
| 73 | \( 1 + 14.4T + 73T^{2} \) |
| 79 | \( 1 + 2.70T + 79T^{2} \) |
| 83 | \( 1 - 16.9T + 83T^{2} \) |
| 89 | \( 1 - 1.52T + 89T^{2} \) |
| 97 | \( 1 + 4T + 97T^{2} \) |
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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
−8.716424920937127424608902795774, −7.62825275732571359821152060969, −6.70647755003476607030460074262, −6.03403930986436437592101397190, −5.58518575015332333207613494251, −4.67505048655027284685722919623, −4.07331552786894921182291248463, −3.27086315745740890875196620470, −2.24724952467315039669351320518, −0.77847757740791034949696327729,
0.77847757740791034949696327729, 2.24724952467315039669351320518, 3.27086315745740890875196620470, 4.07331552786894921182291248463, 4.67505048655027284685722919623, 5.58518575015332333207613494251, 6.03403930986436437592101397190, 6.70647755003476607030460074262, 7.62825275732571359821152060969, 8.716424920937127424608902795774
