| L(s) = 1 | + 2·3-s − 2·4-s − 7-s + 9-s − 4·12-s − 13-s + 4·16-s + 6·17-s − 7·19-s − 2·21-s − 3·23-s − 4·27-s + 2·28-s − 9·29-s + 5·31-s − 2·36-s − 2·37-s − 2·39-s − 6·41-s + 43-s − 3·47-s + 8·48-s + 49-s + 12·51-s + 2·52-s + 9·53-s − 14·57-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 4-s − 0.377·7-s + 1/3·9-s − 1.15·12-s − 0.277·13-s + 16-s + 1.45·17-s − 1.60·19-s − 0.436·21-s − 0.625·23-s − 0.769·27-s + 0.377·28-s − 1.67·29-s + 0.898·31-s − 1/3·36-s − 0.328·37-s − 0.320·39-s − 0.937·41-s + 0.152·43-s − 0.437·47-s + 1.15·48-s + 1/7·49-s + 1.68·51-s + 0.277·52-s + 1.23·53-s − 1.85·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−8.757330732079581772765520931090, −7.978242464812655764046923950162, −7.46586270917006775238312750127, −6.18980064919786182432384163275, −5.44649443474508568649879028268, −4.35319250257652523925819018295, −3.66220700490883632331556632732, −2.89985712041313005804128295915, −1.71501148341393495121690262210, 0,
1.71501148341393495121690262210, 2.89985712041313005804128295915, 3.66220700490883632331556632732, 4.35319250257652523925819018295, 5.44649443474508568649879028268, 6.18980064919786182432384163275, 7.46586270917006775238312750127, 7.978242464812655764046923950162, 8.757330732079581772765520931090
