| L(s) = 1 | + 2·7-s − 4·13-s + 8·19-s + 2·25-s − 8·31-s − 4·37-s + 8·43-s + 3·49-s + 20·61-s + 8·67-s + 28·73-s + 16·79-s − 8·91-s + 28·97-s − 8·103-s − 4·109-s − 10·121-s + 127-s + 131-s + 16·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 1.10·13-s + 1.83·19-s + 2/5·25-s − 1.43·31-s − 0.657·37-s + 1.21·43-s + 3/7·49-s + 2.56·61-s + 0.977·67-s + 3.27·73-s + 1.80·79-s − 0.838·91-s + 2.84·97-s − 0.788·103-s − 0.383·109-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 1.38·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16257024 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16257024 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.389762437\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.389762437\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 166 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−8.437938531464112805791107496509, −8.348488734314970985942208775278, −7.72883618089823188068811705304, −7.64643342647130169451475512509, −7.08174780528008802506868726258, −7.02967654283780374362091187658, −6.52381796521785458764501383141, −5.94249352645876407164604558441, −5.52287869235774521302584694922, −5.18864152735764278912922452376, −5.00177236563068938092178423828, −4.66695692425507481056065782525, −3.81643038173270661291096142040, −3.77318485084019457987627529955, −3.24602207672841430819305528383, −2.63040939553461928200624852827, −2.13605317728546818608979081885, −1.90071567302053279367989911233, −0.960172634962689088870097638310, −0.64701077875785089549676795891,
0.64701077875785089549676795891, 0.960172634962689088870097638310, 1.90071567302053279367989911233, 2.13605317728546818608979081885, 2.63040939553461928200624852827, 3.24602207672841430819305528383, 3.77318485084019457987627529955, 3.81643038173270661291096142040, 4.66695692425507481056065782525, 5.00177236563068938092178423828, 5.18864152735764278912922452376, 5.52287869235774521302584694922, 5.94249352645876407164604558441, 6.52381796521785458764501383141, 7.02967654283780374362091187658, 7.08174780528008802506868726258, 7.64643342647130169451475512509, 7.72883618089823188068811705304, 8.348488734314970985942208775278, 8.437938531464112805791107496509
