| L(s) = 1 | + 3-s + 5-s − 4·7-s + 9-s + 13-s + 15-s + 17-s − 4·21-s + 25-s + 27-s − 2·29-s + 4·31-s − 4·35-s − 2·37-s + 39-s − 6·41-s + 4·43-s + 45-s − 12·47-s + 9·49-s + 51-s + 6·53-s − 8·59-s + 14·61-s − 4·63-s + 65-s + 8·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s + 0.277·13-s + 0.258·15-s + 0.242·17-s − 0.872·21-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 0.718·31-s − 0.676·35-s − 0.328·37-s + 0.160·39-s − 0.937·41-s + 0.609·43-s + 0.149·45-s − 1.75·47-s + 9/7·49-s + 0.140·51-s + 0.824·53-s − 1.04·59-s + 1.79·61-s − 0.503·63-s + 0.124·65-s + 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−15.71320935261539, −14.92657324483650, −14.54770357392735, −13.84861936055064, −13.30994728902892, −13.11735155417805, −12.46522178012630, −11.93617073686104, −11.22895497214300, −10.47329213926477, −9.954654129632559, −9.679553738279485, −9.035281510475094, −8.512089184232008, −7.902141991553612, −7.056336122191476, −6.700811550912110, −6.082075067645963, −5.497940673564590, −4.700369809695193, −3.857068174437814, −3.330408480979493, −2.783100937486995, −2.018869639866922, −1.092626215775272, 0,
1.092626215775272, 2.018869639866922, 2.783100937486995, 3.330408480979493, 3.857068174437814, 4.700369809695193, 5.497940673564590, 6.082075067645963, 6.700811550912110, 7.056336122191476, 7.902141991553612, 8.512089184232008, 9.035281510475094, 9.679553738279485, 9.954654129632559, 10.47329213926477, 11.22895497214300, 11.93617073686104, 12.46522178012630, 13.11735155417805, 13.30994728902892, 13.84861936055064, 14.54770357392735, 14.92657324483650, 15.71320935261539
