| L(s) = 1 | − 2·3-s − 3·9-s − 10·17-s − 19-s − 10·25-s + 14·27-s − 8·31-s − 5·49-s + 20·51-s + 2·57-s − 2·59-s + 28·61-s − 26·67-s − 20·71-s + 18·73-s + 20·75-s + 20·79-s − 4·81-s + 16·93-s − 28·101-s − 12·103-s − 30·107-s − 18·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 9-s − 2.42·17-s − 0.229·19-s − 2·25-s + 2.69·27-s − 1.43·31-s − 5/7·49-s + 2.80·51-s + 0.264·57-s − 0.260·59-s + 3.58·61-s − 3.17·67-s − 2.37·71-s + 2.10·73-s + 2.30·75-s + 2.25·79-s − 4/9·81-s + 1.65·93-s − 2.78·101-s − 1.18·103-s − 2.90·107-s − 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 877952 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 877952 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( 1 + T \) |
| good | 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−7.74613152589584229848027600152, −7.18297506166153508271072468025, −6.72598040565036698033673525843, −6.21150502900434643631796858377, −6.13690005259013972771512187418, −5.41676012615338774687665516924, −5.20164006899702178333342847264, −4.65444054474886627728449087886, −3.93346432196685798564362812764, −3.71573461775705014184176995236, −2.56005012019129152265351485884, −2.44767093282979086637766710358, −1.50567723507471670967120404283, 0, 0,
1.50567723507471670967120404283, 2.44767093282979086637766710358, 2.56005012019129152265351485884, 3.71573461775705014184176995236, 3.93346432196685798564362812764, 4.65444054474886627728449087886, 5.20164006899702178333342847264, 5.41676012615338774687665516924, 6.13690005259013972771512187418, 6.21150502900434643631796858377, 6.72598040565036698033673525843, 7.18297506166153508271072468025, 7.74613152589584229848027600152
