| L(s) = 1 | + 4·2-s + 16·4-s − 25·5-s − 172·7-s + 64·8-s − 100·10-s − 132·11-s − 946·13-s − 688·14-s + 256·16-s + 222·17-s + 500·19-s − 400·20-s − 528·22-s − 3.56e3·23-s + 625·25-s − 3.78e3·26-s − 2.75e3·28-s − 2.19e3·29-s + 2.31e3·31-s + 1.02e3·32-s + 888·34-s + 4.30e3·35-s − 1.12e4·37-s + 2.00e3·38-s − 1.60e3·40-s − 1.24e3·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.32·7-s + 0.353·8-s − 0.316·10-s − 0.328·11-s − 1.55·13-s − 0.938·14-s + 1/4·16-s + 0.186·17-s + 0.317·19-s − 0.223·20-s − 0.232·22-s − 1.40·23-s + 1/5·25-s − 1.09·26-s − 0.663·28-s − 0.483·29-s + 0.432·31-s + 0.176·32-s + 0.131·34-s + 0.593·35-s − 1.35·37-s + 0.224·38-s − 0.158·40-s − 0.115·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + p^{2} T \) |
| good | 7 | \( 1 + 172 T + p^{5} T^{2} \) |
| 11 | \( 1 + 12 p T + p^{5} T^{2} \) |
| 13 | \( 1 + 946 T + p^{5} T^{2} \) |
| 17 | \( 1 - 222 T + p^{5} T^{2} \) |
| 19 | \( 1 - 500 T + p^{5} T^{2} \) |
| 23 | \( 1 + 3564 T + p^{5} T^{2} \) |
| 29 | \( 1 + 2190 T + p^{5} T^{2} \) |
| 31 | \( 1 - 2312 T + p^{5} T^{2} \) |
| 37 | \( 1 + 11242 T + p^{5} T^{2} \) |
| 41 | \( 1 + 1242 T + p^{5} T^{2} \) |
| 43 | \( 1 - 20624 T + p^{5} T^{2} \) |
| 47 | \( 1 + 6588 T + p^{5} T^{2} \) |
| 53 | \( 1 - 21066 T + p^{5} T^{2} \) |
| 59 | \( 1 + 7980 T + p^{5} T^{2} \) |
| 61 | \( 1 - 16622 T + p^{5} T^{2} \) |
| 67 | \( 1 - 1808 T + p^{5} T^{2} \) |
| 71 | \( 1 - 24528 T + p^{5} T^{2} \) |
| 73 | \( 1 - 20474 T + p^{5} T^{2} \) |
| 79 | \( 1 + 46240 T + p^{5} T^{2} \) |
| 83 | \( 1 - 51576 T + p^{5} T^{2} \) |
| 89 | \( 1 - 110310 T + p^{5} T^{2} \) |
| 97 | \( 1 + 78382 T + p^{5} T^{2} \) |
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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
−12.51349603644739329167832288917, −11.97292188347513217166156579550, −10.42344163512576123921356288665, −9.506708288086348414600437235769, −7.76649176349146526240796144848, −6.72285702378481989413902670995, −5.38492094945256836834908142985, −3.89205191509611216643843436991, −2.59291567306222782428210522032, 0,
2.59291567306222782428210522032, 3.89205191509611216643843436991, 5.38492094945256836834908142985, 6.72285702378481989413902670995, 7.76649176349146526240796144848, 9.506708288086348414600437235769, 10.42344163512576123921356288665, 11.97292188347513217166156579550, 12.51349603644739329167832288917
