| L(s) = 1 | + 9-s − 28·49-s − 4·79-s + 9·81-s + 44·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 38·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
| L(s) = 1 | + 1/3·9-s − 4·49-s − 0.450·79-s + 81-s + 4.21·109-s + 2.36·121-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 − 2.92·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.690846586\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.690846586\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T^{2} - 8 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | \( 1 \) |
| 7 | \( ( 1 + p T^{2} )^{4} \) |
| good | 11 | \( ( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2}( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | \( ( 1 + 19 T^{2} + 192 T^{4} + 19 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 29 T^{2} + 552 T^{4} - 29 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - p T^{2} )^{8} \) |
| 23 | \( ( 1 - p T^{2} )^{8} \) |
| 29 | \( ( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2}( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | \( ( 1 - p T^{2} )^{8} \) |
| 37 | \( ( 1 + p T^{2} )^{8} \) |
| 41 | \( ( 1 + p T^{2} )^{8} \) |
| 43 | \( ( 1 + p T^{2} )^{8} \) |
| 47 | \( ( 1 + 31 T^{2} - 1248 T^{4} + 31 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - p T^{2} )^{8} \) |
| 59 | \( ( 1 + p T^{2} )^{8} \) |
| 61 | \( ( 1 - p T^{2} )^{8} \) |
| 67 | \( ( 1 + p T^{2} )^{8} \) |
| 71 | \( ( 1 - 12 T + p T^{2} )^{4}( 1 + 12 T + p T^{2} )^{4} \) |
| 73 | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 + p T^{2} )^{8} \) |
| 97 | \( ( 1 - 149 T^{2} + 12792 T^{4} - 149 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−3.66757602569181712572078832711, −3.65600785624503961426552790590, −3.63709164529683102791959750102, −3.53049229026954668664450339733, −3.45164049322575784038043900802, −3.35567775464401697311589236849, −2.96536866773129546340668282070, −2.95526484758851228212125263782, −2.79016608213254387138747017238, −2.66673151280729270925869123778, −2.63753003990022645523641946341, −2.34037757865854227351120497851, −2.25478298834506379796820564988, −2.25301402954564326316207088234, −2.11829633666807282135139996350, −1.64846512255386177036608411458, −1.58882374083613685118690580596, −1.52332908405175634207205167007, −1.47684176182193464628288871530, −1.41222132808456929696604379093, −1.13502930980800640525874500216, −0.66637453803972082780804410353, −0.52842630230909642203792953747, −0.48082497055179577479366881988, −0.29913405642686855455488720454,
0.29913405642686855455488720454, 0.48082497055179577479366881988, 0.52842630230909642203792953747, 0.66637453803972082780804410353, 1.13502930980800640525874500216, 1.41222132808456929696604379093, 1.47684176182193464628288871530, 1.52332908405175634207205167007, 1.58882374083613685118690580596, 1.64846512255386177036608411458, 2.11829633666807282135139996350, 2.25301402954564326316207088234, 2.25478298834506379796820564988, 2.34037757865854227351120497851, 2.63753003990022645523641946341, 2.66673151280729270925869123778, 2.79016608213254387138747017238, 2.95526484758851228212125263782, 2.96536866773129546340668282070, 3.35567775464401697311589236849, 3.45164049322575784038043900802, 3.53049229026954668664450339733, 3.63709164529683102791959750102, 3.65600785624503961426552790590, 3.66757602569181712572078832711
Plot not available for L-functions of degree greater than 10.
