| L(s) = 1 | + 1.32·3-s − 0.220·5-s + 2.72·7-s − 1.23·9-s − 4.44·11-s + 6.66·13-s − 0.292·15-s + 7.61·19-s + 3.61·21-s − 4.80·23-s − 4.95·25-s − 5.62·27-s − 3.99·29-s + 3.35·31-s − 5.89·33-s − 0.599·35-s + 5.37·37-s + 8.84·39-s − 7.08·41-s + 4.05·43-s + 0.272·45-s + 2.86·47-s + 0.425·49-s + 11.9·53-s + 0.977·55-s + 10.0·57-s + 9.20·59-s + ⋯ |
| L(s) = 1 | + 0.766·3-s − 0.0984·5-s + 1.02·7-s − 0.413·9-s − 1.33·11-s + 1.84·13-s − 0.0754·15-s + 1.74·19-s + 0.789·21-s − 1.00·23-s − 0.990·25-s − 1.08·27-s − 0.741·29-s + 0.602·31-s − 1.02·33-s − 0.101·35-s + 0.883·37-s + 1.41·39-s − 1.10·41-s + 0.617·43-s + 0.0406·45-s + 0.417·47-s + 0.0607·49-s + 1.63·53-s + 0.131·55-s + 1.33·57-s + 1.19·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.013605897\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.013605897\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 1.32T + 3T^{2} \) |
| 5 | \( 1 + 0.220T + 5T^{2} \) |
| 7 | \( 1 - 2.72T + 7T^{2} \) |
| 11 | \( 1 + 4.44T + 11T^{2} \) |
| 13 | \( 1 - 6.66T + 13T^{2} \) |
| 19 | \( 1 - 7.61T + 19T^{2} \) |
| 23 | \( 1 + 4.80T + 23T^{2} \) |
| 29 | \( 1 + 3.99T + 29T^{2} \) |
| 31 | \( 1 - 3.35T + 31T^{2} \) |
| 37 | \( 1 - 5.37T + 37T^{2} \) |
| 41 | \( 1 + 7.08T + 41T^{2} \) |
| 43 | \( 1 - 4.05T + 43T^{2} \) |
| 47 | \( 1 - 2.86T + 47T^{2} \) |
| 53 | \( 1 - 11.9T + 53T^{2} \) |
| 59 | \( 1 - 9.20T + 59T^{2} \) |
| 61 | \( 1 - 8.32T + 61T^{2} \) |
| 67 | \( 1 + 3.81T + 67T^{2} \) |
| 71 | \( 1 - 3.27T + 71T^{2} \) |
| 73 | \( 1 - 7.55T + 73T^{2} \) |
| 79 | \( 1 - 2.42T + 79T^{2} \) |
| 83 | \( 1 - 3.22T + 83T^{2} \) |
| 89 | \( 1 + 2.29T + 89T^{2} \) |
| 97 | \( 1 - 17.7T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.80872695780745850262092550770, −7.43865332576153479035747948577, −6.19186136894544943969651908968, −5.58167391115547273834920369704, −5.10821766827611332744912035301, −3.94761073109534292865482772443, −3.52616650700260156082347428084, −2.57639238595792912672980952726, −1.88126410913453425738411851011, −0.813411620834972393933020495048,
0.813411620834972393933020495048, 1.88126410913453425738411851011, 2.57639238595792912672980952726, 3.52616650700260156082347428084, 3.94761073109534292865482772443, 5.10821766827611332744912035301, 5.58167391115547273834920369704, 6.19186136894544943969651908968, 7.43865332576153479035747948577, 7.80872695780745850262092550770
