| L(s) = 1 | + 8·7-s − 12·13-s − 8·25-s − 32·31-s + 12·37-s + 32·49-s − 32·67-s + 28·73-s − 96·91-s + 12·97-s − 8·103-s + 28·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 72·169-s + 173-s − 64·175-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 3.02·7-s − 3.32·13-s − 8/5·25-s − 5.74·31-s + 1.97·37-s + 32/7·49-s − 3.90·67-s + 3.27·73-s − 10.0·91-s + 1.21·97-s − 0.788·103-s + 2.54·121-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 + 5.53·169-s + 0.0760·173-s − 4.83·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.319379210\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.319379210\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 16 T^{2} + p^{2} T^{4} )( 1 + 16 T^{2} + p^{2} T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 734 T^{4} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 80 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 1918 T^{4} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 5582 T^{4} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 83 | $C_2^3$ | \( 1 - 13294 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 176 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.06247634840820631062050330689, −5.92788064036406419057259081347, −5.78247907050865601705939611314, −5.44614954292912893799679770863, −5.44396993150058345077812775160, −5.10578216836093293154736559604, −4.92665121799502898290279585845, −4.70570195082917290157367408838, −4.68819201053303965944952136585, −4.68102624445576905568846525337, −4.13187257969429050572756992499, −3.87847175613909154757161520135, −3.74603047936285064636462811879, −3.60619632621395287329279319980, −3.38350120180568972042326715239, −2.71163726827378816568677018601, −2.50369773389795496160012895247, −2.45266789053355413575472457655, −2.32125957761453669181526416701, −1.81226128830565253707873920480, −1.69596936132741037757121597360, −1.56689588473057215711326200118, −1.32746091118411572805236413950, −0.42169638626804212821357084229, −0.34594247557794549912419409904,
0.34594247557794549912419409904, 0.42169638626804212821357084229, 1.32746091118411572805236413950, 1.56689588473057215711326200118, 1.69596936132741037757121597360, 1.81226128830565253707873920480, 2.32125957761453669181526416701, 2.45266789053355413575472457655, 2.50369773389795496160012895247, 2.71163726827378816568677018601, 3.38350120180568972042326715239, 3.60619632621395287329279319980, 3.74603047936285064636462811879, 3.87847175613909154757161520135, 4.13187257969429050572756992499, 4.68102624445576905568846525337, 4.68819201053303965944952136585, 4.70570195082917290157367408838, 4.92665121799502898290279585845, 5.10578216836093293154736559604, 5.44396993150058345077812775160, 5.44614954292912893799679770863, 5.78247907050865601705939611314, 5.92788064036406419057259081347, 6.06247634840820631062050330689
