L(s) = 1 | + 5-s + 3·7-s − 3·9-s + 6·11-s − 6·13-s − 17-s − 3·23-s − 4·25-s + 2·29-s + 3·31-s + 3·35-s − 2·37-s − 11·41-s − 6·43-s − 3·45-s − 6·47-s + 2·49-s + 2·53-s + 6·55-s − 6·59-s − 10·61-s − 9·63-s − 6·65-s − 3·67-s + 12·71-s − 9·73-s + 18·77-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 1.13·7-s − 9-s + 1.80·11-s − 1.66·13-s − 0.242·17-s − 0.625·23-s − 4/5·25-s + 0.371·29-s + 0.538·31-s + 0.507·35-s − 0.328·37-s − 1.71·41-s − 0.914·43-s − 0.447·45-s − 0.875·47-s + 2/7·49-s + 0.274·53-s + 0.809·55-s − 0.781·59-s − 1.28·61-s − 1.13·63-s − 0.744·65-s − 0.366·67-s + 1.42·71-s − 1.05·73-s + 2.05·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 251 | \( 1 - T \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−7.56717129674196884140771780785, −6.65347673829557710445358401049, −6.22729162209651997434628428760, −5.24077910615478232015631837364, −4.81031159932706632980948311243, −3.98358589014684508681390601439, −3.03804901519563815964868413467, −2.05959073794528069362740121509, −1.49430408491039579682217369048, 0,
1.49430408491039579682217369048, 2.05959073794528069362740121509, 3.03804901519563815964868413467, 3.98358589014684508681390601439, 4.81031159932706632980948311243, 5.24077910615478232015631837364, 6.22729162209651997434628428760, 6.65347673829557710445358401049, 7.56717129674196884140771780785