| L(s) = 1 | − 9·4-s + 17·16-s − 250·25-s − 900·37-s + 360·43-s + 423·64-s − 1.48e3·67-s − 2.76e3·79-s + 2.25e3·100-s + 108·109-s − 1.96e3·121-s + 127-s + 131-s + 137-s + 139-s + 8.10e3·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4.39e3·169-s − 3.24e3·172-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 9/8·4-s + 0.265·16-s − 2·25-s − 3.99·37-s + 1.27·43-s + 0.826·64-s − 2.69·67-s − 3.94·79-s + 9/4·100-s + 0.0949·109-s − 1.47·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 4.49·148-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 2·169-s − 1.43·172-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + 9 T^{2} + p^{6} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 1962 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 22734 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 21222 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 450 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 180 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 50346 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 740 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 242478 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 1384 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
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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
−10.26837571227162746484887336399, −10.22345143672542539377907321659, −9.359889792137238890421481932832, −9.246863196431183422737521567858, −8.560411296141730710991418894925, −8.502732674819621743434886646713, −7.68185555225961218857103254846, −7.33607592955204318734186367298, −6.81639193963138171549106642308, −6.13706384992762947213800410844, −5.47023633150184299802934924684, −5.37765930889433709071841057276, −4.46406743721194004553333993575, −4.18480204734756348458257590005, −3.58098227001465928934103963452, −2.94871509377801391643707825415, −1.98360564405467700329272522081, −1.38095841218195917023758122098, 0, 0,
1.38095841218195917023758122098, 1.98360564405467700329272522081, 2.94871509377801391643707825415, 3.58098227001465928934103963452, 4.18480204734756348458257590005, 4.46406743721194004553333993575, 5.37765930889433709071841057276, 5.47023633150184299802934924684, 6.13706384992762947213800410844, 6.81639193963138171549106642308, 7.33607592955204318734186367298, 7.68185555225961218857103254846, 8.502732674819621743434886646713, 8.560411296141730710991418894925, 9.246863196431183422737521567858, 9.359889792137238890421481932832, 10.22345143672542539377907321659, 10.26837571227162746484887336399
