| L(s) = 1 | + 4·9-s + 4·25-s − 56·49-s + 16·61-s − 6·81-s + 16·109-s + 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 16·225-s + ⋯ |
| L(s) = 1 | + 4/3·9-s + 4/5·25-s − 8·49-s + 2.04·61-s − 2/3·81-s + 1.53·109-s + 8/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 8/13·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 1.06·225-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.210372873\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.210372873\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 5 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| good | 7 | \( ( 1 + p T^{2} )^{8} \) |
| 11 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 8 T + p T^{2} )^{4}( 1 + 8 T + p T^{2} )^{4} \) |
| 23 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 2 T + p T^{2} )^{8} \) |
| 67 | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 14 T + p T^{2} )^{4}( 1 + 14 T + p T^{2} )^{4} \) |
| 79 | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 - 18 T + p T^{2} )^{4}( 1 + 18 T + p T^{2} )^{4} \) |
| 97 | \( ( 1 - p T^{2} )^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.46499697175697823964427126900, −4.25603133905767588766897207109, −3.98076265085130387070085898363, −3.79069980642322463378539708676, −3.76204759362390188156028090529, −3.73119983478107147034021217367, −3.59887123424009616905464741030, −3.31681626728289923267225395437, −3.13147726395814021045594040500, −3.11158212028217115652841986065, −2.99065885592311772998812773814, −2.89179942929753081499958264419, −2.63088552283868851373844411816, −2.61531076316282649256860049527, −2.33686649266122610980055544545, −1.92873799462557679734214054048, −1.92032048227351568161557389131, −1.76180630575929780086238816138, −1.72895977703801034567059020261, −1.54596618378507389878858806884, −1.34965695404763486023126385774, −1.10590715720127741405859031475, −0.70878724109305976130769868954, −0.56447623842158636909037049944, −0.27104640821909668981851516026,
0.27104640821909668981851516026, 0.56447623842158636909037049944, 0.70878724109305976130769868954, 1.10590715720127741405859031475, 1.34965695404763486023126385774, 1.54596618378507389878858806884, 1.72895977703801034567059020261, 1.76180630575929780086238816138, 1.92032048227351568161557389131, 1.92873799462557679734214054048, 2.33686649266122610980055544545, 2.61531076316282649256860049527, 2.63088552283868851373844411816, 2.89179942929753081499958264419, 2.99065885592311772998812773814, 3.11158212028217115652841986065, 3.13147726395814021045594040500, 3.31681626728289923267225395437, 3.59887123424009616905464741030, 3.73119983478107147034021217367, 3.76204759362390188156028090529, 3.79069980642322463378539708676, 3.98076265085130387070085898363, 4.25603133905767588766897207109, 4.46499697175697823964427126900
Plot not available for L-functions of degree greater than 10.
