| L(s) = 1 | − 2-s + 4-s − 8-s − 2·9-s + 16-s + 2·18-s + 6·25-s + 12·29-s − 32-s − 2·36-s + 4·37-s − 7·49-s − 6·50-s − 4·53-s − 12·58-s + 64-s + 2·72-s − 4·74-s − 5·81-s + 7·98-s + 6·100-s + 4·106-s + 20·109-s − 20·113-s + 12·116-s − 6·121-s + 127-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 2/3·9-s + 1/4·16-s + 0.471·18-s + 6/5·25-s + 2.22·29-s − 0.176·32-s − 1/3·36-s + 0.657·37-s − 49-s − 0.848·50-s − 0.549·53-s − 1.57·58-s + 1/8·64-s + 0.235·72-s − 0.464·74-s − 5/9·81-s + 0.707·98-s + 3/5·100-s + 0.388·106-s + 1.91·109-s − 1.88·113-s + 1.11·116-s − 0.545·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 264992 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264992 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.127868412\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.127868412\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.732513533676001741842395806222, −8.551927308730441194704839251316, −8.063981000805844388953005647957, −7.66704781277271045462237199851, −6.92698888721859570486354553633, −6.66682723299597701313405673479, −6.13700769509257057687309934904, −5.65604719027113190184717005364, −4.88621405812987752749804662581, −4.61728983505646359153586500613, −3.73582709133899820263472460096, −2.88461387934840956020682527470, −2.75784428720712104650553778532, −1.66699130827836424980460714801, −0.73238545477873846723316415123,
0.73238545477873846723316415123, 1.66699130827836424980460714801, 2.75784428720712104650553778532, 2.88461387934840956020682527470, 3.73582709133899820263472460096, 4.61728983505646359153586500613, 4.88621405812987752749804662581, 5.65604719027113190184717005364, 6.13700769509257057687309934904, 6.66682723299597701313405673479, 6.92698888721859570486354553633, 7.66704781277271045462237199851, 8.063981000805844388953005647957, 8.551927308730441194704839251316, 8.732513533676001741842395806222
