| L(s) = 1 | − 7·3-s + 3·5-s − 7·7-s + 22·9-s + 11·11-s − 16·13-s − 21·15-s + 6·17-s − 14·19-s + 49·21-s + 51·23-s − 116·25-s + 35·27-s + 54·29-s − 95·31-s − 77·33-s − 21·35-s − 193·37-s + 112·39-s + 102·41-s − 284·43-s + 66·45-s + 72·47-s + 49·49-s − 42·51-s − 102·53-s + 33·55-s + ⋯ |
| L(s) = 1 | − 1.34·3-s + 0.268·5-s − 0.377·7-s + 0.814·9-s + 0.301·11-s − 0.341·13-s − 0.361·15-s + 0.0856·17-s − 0.169·19-s + 0.509·21-s + 0.462·23-s − 0.927·25-s + 0.249·27-s + 0.345·29-s − 0.550·31-s − 0.406·33-s − 0.101·35-s − 0.857·37-s + 0.459·39-s + 0.388·41-s − 1.00·43-s + 0.218·45-s + 0.223·47-s + 1/7·49-s − 0.115·51-s − 0.264·53-s + 0.0809·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8390369682\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8390369682\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + p T \) |
| 11 | \( 1 - p T \) |
| good | 3 | \( 1 + 7 T + p^{3} T^{2} \) |
| 5 | \( 1 - 3 T + p^{3} T^{2} \) |
| 13 | \( 1 + 16 T + p^{3} T^{2} \) |
| 17 | \( 1 - 6 T + p^{3} T^{2} \) |
| 19 | \( 1 + 14 T + p^{3} T^{2} \) |
| 23 | \( 1 - 51 T + p^{3} T^{2} \) |
| 29 | \( 1 - 54 T + p^{3} T^{2} \) |
| 31 | \( 1 + 95 T + p^{3} T^{2} \) |
| 37 | \( 1 + 193 T + p^{3} T^{2} \) |
| 41 | \( 1 - 102 T + p^{3} T^{2} \) |
| 43 | \( 1 + 284 T + p^{3} T^{2} \) |
| 47 | \( 1 - 72 T + p^{3} T^{2} \) |
| 53 | \( 1 + 102 T + p^{3} T^{2} \) |
| 59 | \( 1 - 63 T + p^{3} T^{2} \) |
| 61 | \( 1 + 790 T + p^{3} T^{2} \) |
| 67 | \( 1 - 433 T + p^{3} T^{2} \) |
| 71 | \( 1 + 135 T + p^{3} T^{2} \) |
| 73 | \( 1 + 238 T + p^{3} T^{2} \) |
| 79 | \( 1 + 770 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1008 T + p^{3} T^{2} \) |
| 89 | \( 1 + 639 T + p^{3} T^{2} \) |
| 97 | \( 1 - 11 T + p^{3} T^{2} \) |
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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
−9.526846379303745021361816811962, −8.623204073132885320184012521200, −7.45996151703375404392657966977, −6.66353884379966107682917517296, −5.97791237612922068045076662424, −5.26022007448378220525193832444, −4.38681045651881128176748978637, −3.18784149417651757968685250173, −1.77705750801008135318994373387, −0.49759877392967676934677195961,
0.49759877392967676934677195961, 1.77705750801008135318994373387, 3.18784149417651757968685250173, 4.38681045651881128176748978637, 5.26022007448378220525193832444, 5.97791237612922068045076662424, 6.66353884379966107682917517296, 7.45996151703375404392657966977, 8.623204073132885320184012521200, 9.526846379303745021361816811962
