| L(s) = 1 | − 2·2-s + 2·3-s + 3·4-s − 2·5-s − 4·6-s + 2·7-s − 4·8-s + 4·10-s + 6·12-s + 8·13-s − 4·14-s − 4·15-s + 5·16-s + 8·17-s + 2·19-s − 6·20-s + 4·21-s + 6·23-s − 8·24-s + 3·25-s − 16·26-s − 2·27-s + 6·28-s + 2·29-s + 8·30-s + 8·31-s − 6·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s − 0.894·5-s − 1.63·6-s + 0.755·7-s − 1.41·8-s + 1.26·10-s + 1.73·12-s + 2.21·13-s − 1.06·14-s − 1.03·15-s + 5/4·16-s + 1.94·17-s + 0.458·19-s − 1.34·20-s + 0.872·21-s + 1.25·23-s − 1.63·24-s + 3/5·25-s − 3.13·26-s − 0.384·27-s + 1.13·28-s + 0.371·29-s + 1.46·30-s + 1.43·31-s − 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71740900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71740900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.011797545\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.011797545\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 36 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2 T - 16 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 72 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 64 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 14 T + 152 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 4 T - 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 16 T + 158 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 2 T + 84 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 16 T + 182 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 134 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2 T + 192 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−7.998230116829811816063117787802, −7.948602980300326526203846525177, −7.30971673960683305591602723172, −7.27458211560247622726114967852, −6.85472159487226386161585787760, −6.31944989851239131294642289650, −5.88869373553919900668073849168, −5.81017465198320629073105775471, −5.15990683524937834056175508296, −4.89671794435961244188185087299, −4.26330379660979631485177717867, −3.83080123405618739694916607871, −3.46777737609915693610429276406, −3.26931317018699918414215610111, −2.75469352950796797406717442898, −2.58948697244380533346049524957, −1.79826922249000786052941595094, −1.33382075137780746362590914466, −0.904022675554635204604662048525, −0.74346325241763926075969472272,
0.74346325241763926075969472272, 0.904022675554635204604662048525, 1.33382075137780746362590914466, 1.79826922249000786052941595094, 2.58948697244380533346049524957, 2.75469352950796797406717442898, 3.26931317018699918414215610111, 3.46777737609915693610429276406, 3.83080123405618739694916607871, 4.26330379660979631485177717867, 4.89671794435961244188185087299, 5.15990683524937834056175508296, 5.81017465198320629073105775471, 5.88869373553919900668073849168, 6.31944989851239131294642289650, 6.85472159487226386161585787760, 7.27458211560247622726114967852, 7.30971673960683305591602723172, 7.948602980300326526203846525177, 7.998230116829811816063117787802
