| L(s) = 1 | + 3-s + 5-s − 2·7-s + 9-s − 2·11-s − 6·13-s + 15-s − 6·17-s + 6·19-s − 2·21-s + 2·23-s + 25-s + 27-s − 2·29-s − 4·31-s − 2·33-s − 2·35-s − 10·37-s − 6·39-s − 2·41-s − 8·43-s + 45-s − 6·47-s − 3·49-s − 6·51-s − 6·53-s − 2·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.603·11-s − 1.66·13-s + 0.258·15-s − 1.45·17-s + 1.37·19-s − 0.436·21-s + 0.417·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s − 0.718·31-s − 0.348·33-s − 0.338·35-s − 1.64·37-s − 0.960·39-s − 0.312·41-s − 1.21·43-s + 0.149·45-s − 0.875·47-s − 3/7·49-s − 0.840·51-s − 0.824·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−8.925096883789773521477287805917, −8.068775843057347650548015775048, −7.04567618337577301272777483456, −6.79209553686319650524661962515, −5.37544434133170083977751837577, −4.90031318356292023726229605539, −3.58207741810454561594516688553, −2.75768606055125216516671650131, −1.91529347433011582491924549952, 0,
1.91529347433011582491924549952, 2.75768606055125216516671650131, 3.58207741810454561594516688553, 4.90031318356292023726229605539, 5.37544434133170083977751837577, 6.79209553686319650524661962515, 7.04567618337577301272777483456, 8.068775843057347650548015775048, 8.925096883789773521477287805917
