| L(s) = 1 | + 3-s + 4·7-s + 9-s + 2·13-s − 2·17-s − 4·19-s + 4·21-s + 27-s + 2·29-s − 10·37-s + 2·39-s − 6·41-s − 43-s − 8·47-s + 9·49-s − 2·51-s − 2·53-s − 4·57-s + 2·61-s + 4·63-s − 4·67-s − 8·71-s − 10·73-s + 8·79-s + 81-s + 16·83-s + 2·87-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.51·7-s + 1/3·9-s + 0.554·13-s − 0.485·17-s − 0.917·19-s + 0.872·21-s + 0.192·27-s + 0.371·29-s − 1.64·37-s + 0.320·39-s − 0.937·41-s − 0.152·43-s − 1.16·47-s + 9/7·49-s − 0.280·51-s − 0.274·53-s − 0.529·57-s + 0.256·61-s + 0.503·63-s − 0.488·67-s − 0.949·71-s − 1.17·73-s + 0.900·79-s + 1/9·81-s + 1.75·83-s + 0.214·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.72956915199536, −14.34074755194000, −13.63943511354978, −13.43686868742567, −12.78888578555483, −12.02852439363532, −11.76666613581042, −11.06484388509005, −10.61500741489169, −10.27626232990258, −9.417231697659047, −8.838834659174726, −8.424881249639304, −8.114969246272401, −7.474630299720612, −6.816086802650330, −6.341528685403014, −5.522978037928955, −4.870759236549725, −4.535725059035371, −3.788810804434057, −3.214976234509637, −2.297872695438829, −1.783263966381482, −1.240574429615346, 0,
1.240574429615346, 1.783263966381482, 2.297872695438829, 3.214976234509637, 3.788810804434057, 4.535725059035371, 4.870759236549725, 5.522978037928955, 6.341528685403014, 6.816086802650330, 7.474630299720612, 8.114969246272401, 8.424881249639304, 8.838834659174726, 9.417231697659047, 10.27626232990258, 10.61500741489169, 11.06484388509005, 11.76666613581042, 12.02852439363532, 12.78888578555483, 13.43686868742567, 13.63943511354978, 14.34074755194000, 14.72956915199536
