| L(s) = 1 | − 4-s + 6·9-s + 8·11-s + 16-s − 12·29-s − 16·31-s − 6·36-s − 4·41-s − 8·44-s − 16·59-s + 28·61-s − 64-s − 32·71-s + 16·79-s + 27·81-s + 20·89-s + 48·99-s + 12·101-s − 12·109-s + 12·116-s + 26·121-s + 16·124-s + 127-s + 131-s + 137-s + 139-s + 6·144-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 2·9-s + 2.41·11-s + 1/4·16-s − 2.22·29-s − 2.87·31-s − 36-s − 0.624·41-s − 1.20·44-s − 2.08·59-s + 3.58·61-s − 1/8·64-s − 3.79·71-s + 1.80·79-s + 3·81-s + 2.11·89-s + 4.82·99-s + 1.19·101-s − 1.14·109-s + 1.11·116-s + 2.36·121-s + 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1/2·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.865843463\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.865843463\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−9.268353699802693065051488763483, −8.989726762343082665394647968083, −8.474702556584352627174212379876, −7.88699348270199839331785260553, −7.36346163818633471784470462890, −7.35723255013866401975578666122, −6.75455176798058391188704523467, −6.70621357434304784754136437266, −5.93812612562973689655658734544, −5.72049455523012331634926904893, −5.15400445526562305421510023393, −4.66023702078066563583429189332, −4.14732215425750944075815496081, −4.01466257195231029512624797866, −3.46279013586838455807860304301, −3.40127290978770289913697773548, −1.90579381612495718147824086067, −1.90173568409539422613977127687, −1.39990125129625124263851426966, −0.60619432205379762246235719692,
0.60619432205379762246235719692, 1.39990125129625124263851426966, 1.90173568409539422613977127687, 1.90579381612495718147824086067, 3.40127290978770289913697773548, 3.46279013586838455807860304301, 4.01466257195231029512624797866, 4.14732215425750944075815496081, 4.66023702078066563583429189332, 5.15400445526562305421510023393, 5.72049455523012331634926904893, 5.93812612562973689655658734544, 6.70621357434304784754136437266, 6.75455176798058391188704523467, 7.35723255013866401975578666122, 7.36346163818633471784470462890, 7.88699348270199839331785260553, 8.474702556584352627174212379876, 8.989726762343082665394647968083, 9.268353699802693065051488763483
