| L(s) = 1 | + 3-s + 5-s + 4·7-s + 3·11-s + 10·13-s + 15-s + 5·19-s + 4·21-s − 9·23-s − 27-s − 10·31-s + 3·33-s + 4·35-s + 37-s + 10·39-s + 18·41-s − 16·43-s + 3·47-s + 9·49-s + 3·53-s + 3·55-s + 5·57-s + 12·59-s − 8·61-s + 10·65-s + 8·67-s − 9·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.51·7-s + 0.904·11-s + 2.77·13-s + 0.258·15-s + 1.14·19-s + 0.872·21-s − 1.87·23-s − 0.192·27-s − 1.79·31-s + 0.522·33-s + 0.676·35-s + 0.164·37-s + 1.60·39-s + 2.81·41-s − 2.43·43-s + 0.437·47-s + 9/7·49-s + 0.412·53-s + 0.404·55-s + 0.662·57-s + 1.56·59-s − 1.02·61-s + 1.24·65-s + 0.977·67-s − 1.08·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2822400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2822400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.445693254\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.445693254\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 10 T + 69 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 3 T - 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 8 T - 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.309568326150923309846155751601, −9.040056350434775883628627772224, −8.877724567996328616021617136622, −8.296663134525459770718527743765, −7.988940131668263402720202880113, −7.85015411825638768282746229982, −7.25974767269959854568100372988, −6.70323137814021044374800571064, −6.24169142991915731603179295077, −5.94960941262138441334640599513, −5.45040416650729767509343950410, −5.23405574975122137915420468218, −4.40990571025039378676722920364, −3.95383269126871960956118874879, −3.70910739761010372983426523772, −3.35349640486529262565070461487, −2.36544985645178712678795802340, −1.94719192154439855893068828703, −1.39142699118696885143215868270, −0.997541179453553047899369005564,
0.997541179453553047899369005564, 1.39142699118696885143215868270, 1.94719192154439855893068828703, 2.36544985645178712678795802340, 3.35349640486529262565070461487, 3.70910739761010372983426523772, 3.95383269126871960956118874879, 4.40990571025039378676722920364, 5.23405574975122137915420468218, 5.45040416650729767509343950410, 5.94960941262138441334640599513, 6.24169142991915731603179295077, 6.70323137814021044374800571064, 7.25974767269959854568100372988, 7.85015411825638768282746229982, 7.988940131668263402720202880113, 8.296663134525459770718527743765, 8.877724567996328616021617136622, 9.040056350434775883628627772224, 9.309568326150923309846155751601
