| L(s) = 1 | − 1.43e7·9-s − 7.95e8·13-s + 6.10e10·25-s + 2.18e12·37-s − 7.94e12·49-s + 8.04e13·61-s − 1.80e13·73-s + 2.05e14·81-s − 2.07e15·97-s + 7.56e15·109-s + 1.14e16·117-s − 8.35e15·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3.72e17·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 9-s − 3.51·13-s + 2·25-s + 3.77·37-s − 1.67·49-s + 3.27·61-s − 0.191·73-s + 81-s − 2.60·97-s + 3.96·109-s + 3.51·117-s − 2·121-s + 7.27·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+15/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(1.545762059\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.545762059\) |
| \(L(\frac{17}{2})\) |
|
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 | $C_2$ | \( 1 + p^{15} T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - p^{15} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 1244900 T + p^{15} T^{2} )( 1 + 1244900 T + p^{15} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p^{15} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 397771850 T + p^{15} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p^{15} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 7700827736 T + p^{15} T^{2} )( 1 + 7700827736 T + p^{15} T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p^{15} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p^{15} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 213681227452 T + p^{15} T^{2} )( 1 + 213681227452 T + p^{15} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 1090158909950 T + p^{15} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p^{15} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 1440654152600 T + p^{15} T^{2} )( 1 + 1440654152600 T + p^{15} T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p^{15} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p^{15} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{15} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 40241378988902 T + p^{15} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 99059017336400 T + p^{15} T^{2} )( 1 + 99059017336400 T + p^{15} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p^{15} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 9014812804550 T + p^{15} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 88692309079036 T + p^{15} T^{2} )( 1 + 88692309079036 T + p^{15} T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p^{15} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p^{15} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 1035097921427150 T + p^{15} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.80029724288571445459555573873, −12.23522215755240141454800418769, −11.47734321015336108536946121553, −11.27486487334117969209198892514, −10.30234657975409394652183855243, −9.611557338147145085935086698459, −9.561274458168026942426641653558, −8.556019123275750066582344319012, −7.942138671394704441794780597555, −7.35527813110555066631946417986, −6.81506065607060976759034327428, −6.05066147775036666564772688306, −5.05624127637084342098669304112, −4.97219392236716757940367791113, −4.16535155891509293439985709959, −2.88154518625894721560154918976, −2.70035417282480657590549531731, −2.14636098093106213231300476566, −0.882929558798614464491835688678, −0.37199257285123556888560748392,
0.37199257285123556888560748392, 0.882929558798614464491835688678, 2.14636098093106213231300476566, 2.70035417282480657590549531731, 2.88154518625894721560154918976, 4.16535155891509293439985709959, 4.97219392236716757940367791113, 5.05624127637084342098669304112, 6.05066147775036666564772688306, 6.81506065607060976759034327428, 7.35527813110555066631946417986, 7.942138671394704441794780597555, 8.556019123275750066582344319012, 9.561274458168026942426641653558, 9.611557338147145085935086698459, 10.30234657975409394652183855243, 11.27486487334117969209198892514, 11.47734321015336108536946121553, 12.23522215755240141454800418769, 12.80029724288571445459555573873
