| L(s) = 1 | + 4·4-s + 5-s + 11·11-s + 16·16-s + 4·20-s − 35·23-s − 24·25-s − 37·31-s − 25·37-s + 44·44-s − 50·47-s + 49·49-s + 70·53-s + 11·55-s − 107·59-s + 64·64-s + 35·67-s + 133·71-s + 16·80-s + 97·89-s − 140·92-s + 95·97-s − 96·100-s − 190·103-s − 215·113-s − 35·115-s + ⋯ |
| L(s) = 1 | + 4-s + 1/5·5-s + 11-s + 16-s + 1/5·20-s − 1.52·23-s − 0.959·25-s − 1.19·31-s − 0.675·37-s + 44-s − 1.06·47-s + 49-s + 1.32·53-s + 1/5·55-s − 1.81·59-s + 64-s + 0.522·67-s + 1.87·71-s + 1/5·80-s + 1.08·89-s − 1.52·92-s + 0.979·97-s − 0.959·100-s − 1.84·103-s − 1.90·113-s − 0.304·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.663244662\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.663244662\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - p T \) |
| good | 2 | \( ( 1 - p T )( 1 + p T ) \) |
| 5 | \( 1 - T + p^{2} T^{2} \) |
| 7 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( ( 1 - p T )( 1 + p T ) \) |
| 17 | \( ( 1 - p T )( 1 + p T ) \) |
| 19 | \( ( 1 - p T )( 1 + p T ) \) |
| 23 | \( 1 + 35 T + p^{2} T^{2} \) |
| 29 | \( ( 1 - p T )( 1 + p T ) \) |
| 31 | \( 1 + 37 T + p^{2} T^{2} \) |
| 37 | \( 1 + 25 T + p^{2} T^{2} \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( ( 1 - p T )( 1 + p T ) \) |
| 47 | \( 1 + 50 T + p^{2} T^{2} \) |
| 53 | \( 1 - 70 T + p^{2} T^{2} \) |
| 59 | \( 1 + 107 T + p^{2} T^{2} \) |
| 61 | \( ( 1 - p T )( 1 + p T ) \) |
| 67 | \( 1 - 35 T + p^{2} T^{2} \) |
| 71 | \( 1 - 133 T + p^{2} T^{2} \) |
| 73 | \( ( 1 - p T )( 1 + p T ) \) |
| 79 | \( ( 1 - p T )( 1 + p T ) \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( 1 - 97 T + p^{2} T^{2} \) |
| 97 | \( 1 - 95 T + p^{2} T^{2} \) |
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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
−13.77308283417076469757182569636, −12.34276895270773157406270356238, −11.63158093484016028018114709727, −10.52642079345654643617860241017, −9.414479021268497706432515173756, −7.963407842995685997600300109804, −6.76996973244600300009168836210, −5.74200427873120567136072829790, −3.75639514250106391357781493954, −1.91757642055280182702741978380,
1.91757642055280182702741978380, 3.75639514250106391357781493954, 5.74200427873120567136072829790, 6.76996973244600300009168836210, 7.963407842995685997600300109804, 9.414479021268497706432515173756, 10.52642079345654643617860241017, 11.63158093484016028018114709727, 12.34276895270773157406270356238, 13.77308283417076469757182569636
