| L(s) = 1 | − 3-s + 4·7-s + 9-s − 11-s + 13-s + 17-s − 2·19-s − 4·21-s − 5·25-s − 27-s − 6·29-s − 8·31-s + 33-s + 2·37-s − 39-s − 6·41-s − 2·43-s + 6·47-s + 9·49-s − 51-s + 6·53-s + 2·57-s − 6·59-s − 4·61-s + 4·63-s + 16·67-s − 12·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.51·7-s + 1/3·9-s − 0.301·11-s + 0.277·13-s + 0.242·17-s − 0.458·19-s − 0.872·21-s − 25-s − 0.192·27-s − 1.11·29-s − 1.43·31-s + 0.174·33-s + 0.328·37-s − 0.160·39-s − 0.937·41-s − 0.304·43-s + 0.875·47-s + 9/7·49-s − 0.140·51-s + 0.824·53-s + 0.264·57-s − 0.781·59-s − 0.512·61-s + 0.503·63-s + 1.95·67-s − 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116688 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.653149733\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.653149733\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.59187223037512, −13.08232262929124, −12.63829444011888, −11.97416389886834, −11.58538199087592, −11.27656910626860, −10.65613866297091, −10.45026987015630, −9.743852781425383, −9.084650366370629, −8.724023692912357, −7.994528983194775, −7.693532821644928, −7.245614733048098, −6.571887233660601, −5.842203912505412, −5.526647888896019, −5.041476915462246, −4.444390782405623, −3.914468684030791, −3.373435452400912, −2.318793337805726, −1.862520204761349, −1.344441776227013, −0.4094341178860706,
0.4094341178860706, 1.344441776227013, 1.862520204761349, 2.318793337805726, 3.373435452400912, 3.914468684030791, 4.444390782405623, 5.041476915462246, 5.526647888896019, 5.842203912505412, 6.571887233660601, 7.245614733048098, 7.693532821644928, 7.994528983194775, 8.724023692912357, 9.084650366370629, 9.743852781425383, 10.45026987015630, 10.65613866297091, 11.27656910626860, 11.58538199087592, 11.97416389886834, 12.63829444011888, 13.08232262929124, 13.59187223037512
