| L(s) = 1 | + 2·5-s + 6·13-s + 2·17-s − 25-s − 10·29-s + 2·37-s + 10·41-s + 14·53-s − 10·61-s + 12·65-s + 6·73-s + 4·85-s + 10·89-s − 18·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.66·13-s + 0.485·17-s − 1/5·25-s − 1.85·29-s + 0.328·37-s + 1.56·41-s + 1.92·53-s − 1.28·61-s + 1.48·65-s + 0.702·73-s + 0.433·85-s + 1.05·89-s − 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.236738115\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.236738115\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−15.23126924231414, −14.62231496912688, −14.09679812534119, −13.50337282594002, −13.25354843564355, −12.70818368684145, −11.97817696371446, −11.39954178893236, −10.79502826037676, −10.51648825892562, −9.591241470551346, −9.369584703390059, −8.740789350270046, −8.084171819608833, −7.517105945781675, −6.831212246018006, −6.008962910982877, −5.838833411838287, −5.262115163700634, −4.215381522959634, −3.782686032082528, −3.017936623689607, −2.148963938570139, −1.537298008944576, −0.7242988573788277,
0.7242988573788277, 1.537298008944576, 2.148963938570139, 3.017936623689607, 3.782686032082528, 4.215381522959634, 5.262115163700634, 5.838833411838287, 6.008962910982877, 6.831212246018006, 7.517105945781675, 8.084171819608833, 8.740789350270046, 9.369584703390059, 9.591241470551346, 10.51648825892562, 10.79502826037676, 11.39954178893236, 11.97817696371446, 12.70818368684145, 13.25354843564355, 13.50337282594002, 14.09679812534119, 14.62231496912688, 15.23126924231414
