| L(s) = 1 | − 16·31-s + 56·61-s − 9·81-s + 44·121-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 + ⋯ |
| L(s) = 1 | − 2.87·31-s + 7.17·61-s − 81-s + 4·121-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 + 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 + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + 0.0644·241-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.087427694\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.087427694\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^3$ | \( 1 - 94 T^{4} + p^{4} T^{8} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 13 | $C_2^3$ | \( 1 + 146 T^{4} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 37 | $C_2^3$ | \( 1 - 2062 T^{4} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 43 | $C_2^3$ | \( 1 - 3214 T^{4} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{4} \) |
| 67 | $C_2^3$ | \( 1 + 5906 T^{4} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 - 8542 T^{4} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $C_2^3$ | \( 1 - 18814 T^{4} + p^{4} T^{8} \) |
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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
−7.00605239952626062650900349976, −6.90226927090688412518711403268, −6.46906986043415664692664949963, −6.33360383097670384700274976645, −6.14637066227117834742440675677, −5.69729957882976061251911394362, −5.62640547881428672876678861360, −5.47592838312825359066453933285, −5.19937805363465350268584380509, −4.95108372630036333510499726638, −4.83530600642989268922747141246, −4.48676846074074524970299303122, −3.97674849085225271863939745011, −3.81545736973331554632601859933, −3.78715910760486859992191564319, −3.72538329168478721571651129590, −3.06408519491111952680578615938, −2.97525660020543560451748516251, −2.58936030361621522850748381655, −2.14241736159813608414000170484, −1.97157040146593949985367041054, −1.91299801198480199484279969546, −1.16354033796269817599740498782, −0.894976030896496959444208243933, −0.34629963828253387581457182745,
0.34629963828253387581457182745, 0.894976030896496959444208243933, 1.16354033796269817599740498782, 1.91299801198480199484279969546, 1.97157040146593949985367041054, 2.14241736159813608414000170484, 2.58936030361621522850748381655, 2.97525660020543560451748516251, 3.06408519491111952680578615938, 3.72538329168478721571651129590, 3.78715910760486859992191564319, 3.81545736973331554632601859933, 3.97674849085225271863939745011, 4.48676846074074524970299303122, 4.83530600642989268922747141246, 4.95108372630036333510499726638, 5.19937805363465350268584380509, 5.47592838312825359066453933285, 5.62640547881428672876678861360, 5.69729957882976061251911394362, 6.14637066227117834742440675677, 6.33360383097670384700274976645, 6.46906986043415664692664949963, 6.90226927090688412518711403268, 7.00605239952626062650900349976
