| L(s) = 1 | − 14·9-s − 44·17-s − 116·41-s + 188·49-s − 140·73-s − 15·81-s + 260·89-s − 536·97-s − 452·113-s − 462·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 616·153-s + 157-s + 163-s + 167-s + 588·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | − 1.55·9-s − 2.58·17-s − 2.82·41-s + 3.83·49-s − 1.91·73-s − 0.185·81-s + 2.92·89-s − 5.52·97-s − 4·113-s − 3.81·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 4.02·153-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 3.47·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4180126439\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4180126439\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( ( 1 + 7 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 21 p T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 294 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 11 T + p^{2} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + 183 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 914 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 474 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 1346 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 582 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 29 T + p^{2} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 2114 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 1054 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 1218 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 2562 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 3878 T^{2} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 7119 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 8926 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 35 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 382 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 13239 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 65 T + p^{2} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 134 T + p^{2} T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.58522953689873805240787319054, −6.36648901812613123397074508191, −6.19938223870904196426240570979, −5.74247987668067692573756217483, −5.49467307544627410526836507783, −5.45936593168005838454700625198, −5.38204984321273140514809656018, −5.07770833244050358081518962385, −4.73019906660257208875097211188, −4.41775772795278570368198763907, −4.39212964428684041375911790937, −3.91518998587487835194305026861, −3.89351701935564008856856089159, −3.70358130169274344243785297509, −3.32252161740540758357830298271, −2.85163092059652189219806268296, −2.66004054910668505215562061555, −2.60026177716307375906943183755, −2.43214020577481015708775222429, −2.04068850235920198592283353532, −1.60048249379661438479869588762, −1.32262165695454133718481736034, −1.09657885370881272999004404132, −0.24188853755585627171663539099, −0.20965251868994067665755197663,
0.20965251868994067665755197663, 0.24188853755585627171663539099, 1.09657885370881272999004404132, 1.32262165695454133718481736034, 1.60048249379661438479869588762, 2.04068850235920198592283353532, 2.43214020577481015708775222429, 2.60026177716307375906943183755, 2.66004054910668505215562061555, 2.85163092059652189219806268296, 3.32252161740540758357830298271, 3.70358130169274344243785297509, 3.89351701935564008856856089159, 3.91518998587487835194305026861, 4.39212964428684041375911790937, 4.41775772795278570368198763907, 4.73019906660257208875097211188, 5.07770833244050358081518962385, 5.38204984321273140514809656018, 5.45936593168005838454700625198, 5.49467307544627410526836507783, 5.74247987668067692573756217483, 6.19938223870904196426240570979, 6.36648901812613123397074508191, 6.58522953689873805240787319054
