| L(s) = 1 | − 2·16-s − 32·67-s + 80·101-s + 64·103-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 88·169-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 + ⋯ |
| L(s) = 1 | − 1/2·16-s − 3.90·67-s + 7.96·101-s + 6.30·103-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 + 6.76·169-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 + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.475474500\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.475474500\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( ( 1 + T^{4} + p^{4} T^{8} )^{2} \) |
| 3 | \( ( 1 + p^{4} T^{8} )^{2} \) |
| 5 | \( ( 1 - 48 T^{4} + p^{4} T^{8} )( 1 + 48 T^{4} + p^{4} T^{8} ) \) |
| 7 | \( 1 + 4034 T^{8} + p^{8} T^{16} \) |
| 11 | \( ( 1 + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 238 T^{4} + p^{4} T^{8} )^{2} \) |
| 23 | \( 1 - 534718 T^{8} + p^{8} T^{16} \) |
| 29 | \( 1 + 20642 T^{8} + p^{8} T^{16} \) |
| 31 | \( 1 - 1809406 T^{8} + p^{8} T^{16} \) |
| 37 | \( ( 1 - 1680 T^{4} + p^{4} T^{8} )( 1 + 1680 T^{4} + p^{4} T^{8} ) \) |
| 41 | \( ( 1 - 96 T^{2} + 4608 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8} )( 1 + 96 T^{2} + 4608 T^{4} + 96 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 43 | \( ( 1 + 1202 T^{4} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - p T^{2} )^{8} \) |
| 53 | \( ( 1 - 718 T^{4} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 6286 T^{4} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 2640 T^{4} + p^{4} T^{8} )( 1 + 2640 T^{4} + p^{4} T^{8} ) \) |
| 67 | \( ( 1 + 4 T + p T^{2} )^{8} \) |
| 71 | \( 1 - 17252926 T^{8} + p^{8} T^{16} \) |
| 73 | \( ( 1 - 10560 T^{4} + p^{4} T^{8} )( 1 + 10560 T^{4} + p^{4} T^{8} ) \) |
| 79 | \( 1 + 73045634 T^{8} + p^{8} T^{16} \) |
| 83 | \( ( 1 + 8722 T^{4} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 16 T + p T^{2} )^{4}( 1 + 16 T + p T^{2} )^{4} \) |
| 97 | \( 1 + 121608962 T^{8} + p^{8} T^{16} \) |
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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
−5.16773885525443689040479838632, −5.10501131582974860702701787541, −5.04703290419720831473774142565, −4.73891676918031591053882963165, −4.65676970058787066027116398630, −4.58272573813362167647593560266, −4.54030010702666757945618285780, −4.27626218335861439146510970434, −4.21639436182380443339416007427, −3.80299289653462403240649702160, −3.66768610800993715881367741692, −3.49930018542454127824634617179, −3.45210746767560870796620410822, −3.20367609514948699255518128899, −3.14326891294711734362242683262, −2.91001708898868259113792935247, −2.62910998232784470428584633312, −2.32441696413495850170768370136, −2.18656454319720471147676389240, −2.16933983264778584539643925041, −1.70004287464726898479489097685, −1.61611817845193482616299093069, −1.32131578152751341104153425809, −0.62497419398261959914472369781, −0.61489538594685549901414893474,
0.61489538594685549901414893474, 0.62497419398261959914472369781, 1.32131578152751341104153425809, 1.61611817845193482616299093069, 1.70004287464726898479489097685, 2.16933983264778584539643925041, 2.18656454319720471147676389240, 2.32441696413495850170768370136, 2.62910998232784470428584633312, 2.91001708898868259113792935247, 3.14326891294711734362242683262, 3.20367609514948699255518128899, 3.45210746767560870796620410822, 3.49930018542454127824634617179, 3.66768610800993715881367741692, 3.80299289653462403240649702160, 4.21639436182380443339416007427, 4.27626218335861439146510970434, 4.54030010702666757945618285780, 4.58272573813362167647593560266, 4.65676970058787066027116398630, 4.73891676918031591053882963165, 5.04703290419720831473774142565, 5.10501131582974860702701787541, 5.16773885525443689040479838632
Plot not available for L-functions of degree greater than 10.
