| L(s) = 1 | − 208·17-s − 234·25-s + 944·41-s − 686·49-s + 2.19e3·73-s + 352·89-s + 1.18e3·97-s − 2.65e3·113-s + 2.66e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4.07e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 2.96·17-s − 1.87·25-s + 3.59·41-s − 2·49-s + 3.52·73-s + 0.419·89-s + 1.24·97-s − 2.21·113-s + 2·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 1.85·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7468019542\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7468019542\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p^{3} T^{2} )( 1 + 4 T + p^{3} T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 18 T + p^{3} T^{2} )( 1 + 18 T + p^{3} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 104 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 284 T + p^{3} T^{2} )( 1 + 284 T + p^{3} T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 214 T + p^{3} T^{2} )( 1 + 214 T + p^{3} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 472 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 572 T + p^{3} T^{2} )( 1 + 572 T + p^{3} T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 830 T + p^{3} T^{2} )( 1 + 830 T + p^{3} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 1098 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 176 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 594 T + p^{3} T^{2} )^{2} \) |
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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
−9.909413659314296968398052073323, −9.318558783853312381684583871259, −8.970006115422341557208766988141, −8.209041411040670357141517541946, −8.155232807849429526434579756431, −7.56244373268233956676761075543, −7.18321009563188328316342222284, −6.61978373304166425076575475314, −6.16727916950176754144370955983, −6.12461171988710701978683178881, −5.35741314783722304954906670220, −4.70755978975232273079404152065, −4.56623665985779360221896090138, −3.78605295307496528134774519253, −3.75637651333367935481440153593, −2.63577257335708424965134835131, −2.32124045454665610269267548388, −1.93899500077636139424083535611, −1.03343237334104771516633581593, −0.22373686823321245993206599400,
0.22373686823321245993206599400, 1.03343237334104771516633581593, 1.93899500077636139424083535611, 2.32124045454665610269267548388, 2.63577257335708424965134835131, 3.75637651333367935481440153593, 3.78605295307496528134774519253, 4.56623665985779360221896090138, 4.70755978975232273079404152065, 5.35741314783722304954906670220, 6.12461171988710701978683178881, 6.16727916950176754144370955983, 6.61978373304166425076575475314, 7.18321009563188328316342222284, 7.56244373268233956676761075543, 8.155232807849429526434579756431, 8.209041411040670357141517541946, 8.970006115422341557208766988141, 9.318558783853312381684583871259, 9.909413659314296968398052073323
