L(s) = 1 | − 3·3-s − 5-s + 6·9-s + 5·11-s + 5·13-s + 3·15-s + 7·17-s − 2·19-s + 2·23-s + 25-s − 9·27-s + 7·29-s + 4·31-s − 15·33-s − 6·37-s − 15·39-s + 12·41-s + 2·43-s − 6·45-s + 47-s − 21·51-s − 5·55-s + 6·57-s − 4·59-s − 4·61-s − 5·65-s − 8·67-s + ⋯ |
L(s) = 1 | − 1.73·3-s − 0.447·5-s + 2·9-s + 1.50·11-s + 1.38·13-s + 0.774·15-s + 1.69·17-s − 0.458·19-s + 0.417·23-s + 1/5·25-s − 1.73·27-s + 1.29·29-s + 0.718·31-s − 2.61·33-s − 0.986·37-s − 2.40·39-s + 1.87·41-s + 0.304·43-s − 0.894·45-s + 0.145·47-s − 2.94·51-s − 0.674·55-s + 0.794·57-s − 0.520·59-s − 0.512·61-s − 0.620·65-s − 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.306903930\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.306903930\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−8.469593510481410672158422929153, −7.53865014479941364859385861433, −6.76183869331890959102062964394, −6.14472686938251363416766925909, −5.73827754277642739938159369313, −4.68034948049666199716320051545, −4.09045006533871894101168233996, −3.23936592321491102560843324681, −1.34814974495374732009706645550, −0.867865270599308918807940906939,
0.867865270599308918807940906939, 1.34814974495374732009706645550, 3.23936592321491102560843324681, 4.09045006533871894101168233996, 4.68034948049666199716320051545, 5.73827754277642739938159369313, 6.14472686938251363416766925909, 6.76183869331890959102062964394, 7.53865014479941364859385861433, 8.469593510481410672158422929153