| L(s) = 1 | + 184·31-s + 296·61-s − 484·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + ⋯ |
| L(s) = 1 | + 5.93·31-s + 4.85·61-s − 4·121-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 + 0.00636·157-s + 0.00613·163-s + 0.00598·167-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 + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + 0.00440·227-s + 0.00436·229-s + 0.00429·233-s + 0.00418·239-s + 0.00414·241-s + 0.00398·251-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(5.609762658\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.609762658\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^3$ | \( 1 + 4034 T^{4} + p^{8} T^{8} \) |
| 11 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 13 | $C_2^3$ | \( 1 - 35806 T^{4} + p^{8} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + p^{4} T^{4} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 31 | $C_2$ | \( ( 1 - 46 T + p^{2} T^{2} )^{4} \) |
| 37 | $C_2^3$ | \( 1 + 503522 T^{4} + p^{8} T^{8} \) |
| 41 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 43 | $C_2^3$ | \( 1 + 3492194 T^{4} + p^{8} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + p^{4} T^{4} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 61 | $C_2$ | \( ( 1 - 74 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^3$ | \( 1 - 5421406 T^{4} + p^{8} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 + 16169282 T^{4} + p^{8} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 7682 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + p^{4} T^{4} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 97 | $C_2^3$ | \( 1 + 176908034 T^{4} + p^{8} T^{8} \) |
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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
−7.12442683250916585350741630207, −6.70624121591675204212595273058, −6.61813048326250707866231021230, −6.43258148004286775186212401563, −6.21918074871280658353886635860, −5.90549985841461974267805116601, −5.58544857845981424724714210737, −5.56645042970157539328072559409, −5.08594214117884969554658000570, −4.95676927869165579185291317111, −4.50355708237636304858609504057, −4.46720611373852692352461655458, −4.46174218951482174865328142362, −3.79328135650175804894968792440, −3.66956336172162189192948292771, −3.51559003293538094290935847549, −2.95196050495942844086382577219, −2.66922462444759398287723210349, −2.54878878664381889802503182251, −2.40368506156744287379014372382, −1.90290192340603316414912275809, −1.35844598773133422633563236109, −0.996885638470349676740755361579, −0.826086511852680433308814628223, −0.41330860879752054591181177297,
0.41330860879752054591181177297, 0.826086511852680433308814628223, 0.996885638470349676740755361579, 1.35844598773133422633563236109, 1.90290192340603316414912275809, 2.40368506156744287379014372382, 2.54878878664381889802503182251, 2.66922462444759398287723210349, 2.95196050495942844086382577219, 3.51559003293538094290935847549, 3.66956336172162189192948292771, 3.79328135650175804894968792440, 4.46174218951482174865328142362, 4.46720611373852692352461655458, 4.50355708237636304858609504057, 4.95676927869165579185291317111, 5.08594214117884969554658000570, 5.56645042970157539328072559409, 5.58544857845981424724714210737, 5.90549985841461974267805116601, 6.21918074871280658353886635860, 6.43258148004286775186212401563, 6.61813048326250707866231021230, 6.70624121591675204212595273058, 7.12442683250916585350741630207
