| L(s) = 1 | + 3-s − 1.04·5-s − 1.46·7-s + 9-s + 2.04·11-s − 6.89·13-s − 1.04·15-s + 7.77·17-s − 6.37·19-s − 1.46·21-s − 3.90·25-s + 27-s + 4.14·29-s + 0.909·31-s + 2.04·33-s + 1.53·35-s − 4.07·37-s − 6.89·39-s + 4.89·41-s + 3.50·43-s − 1.04·45-s − 1.92·47-s − 4.85·49-s + 7.77·51-s + 7.82·53-s − 2.14·55-s − 6.37·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.468·5-s − 0.552·7-s + 0.333·9-s + 0.617·11-s − 1.91·13-s − 0.270·15-s + 1.88·17-s − 1.46·19-s − 0.319·21-s − 0.780·25-s + 0.192·27-s + 0.770·29-s + 0.163·31-s + 0.356·33-s + 0.258·35-s − 0.670·37-s − 1.10·39-s + 0.764·41-s + 0.534·43-s − 0.156·45-s − 0.281·47-s − 0.694·49-s + 1.08·51-s + 1.07·53-s − 0.289·55-s − 0.844·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6348 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6348 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.764953797\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.764953797\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + 1.04T + 5T^{2} \) |
| 7 | \( 1 + 1.46T + 7T^{2} \) |
| 11 | \( 1 - 2.04T + 11T^{2} \) |
| 13 | \( 1 + 6.89T + 13T^{2} \) |
| 17 | \( 1 - 7.77T + 17T^{2} \) |
| 19 | \( 1 + 6.37T + 19T^{2} \) |
| 29 | \( 1 - 4.14T + 29T^{2} \) |
| 31 | \( 1 - 0.909T + 31T^{2} \) |
| 37 | \( 1 + 4.07T + 37T^{2} \) |
| 41 | \( 1 - 4.89T + 41T^{2} \) |
| 43 | \( 1 - 3.50T + 43T^{2} \) |
| 47 | \( 1 + 1.92T + 47T^{2} \) |
| 53 | \( 1 - 7.82T + 53T^{2} \) |
| 59 | \( 1 - 14.4T + 59T^{2} \) |
| 61 | \( 1 - 7.53T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 + 0.197T + 71T^{2} \) |
| 73 | \( 1 + 7.96T + 73T^{2} \) |
| 79 | \( 1 - 4.62T + 79T^{2} \) |
| 83 | \( 1 + 8.28T + 83T^{2} \) |
| 89 | \( 1 - 0.275T + 89T^{2} \) |
| 97 | \( 1 - 7.35T + 97T^{2} \) |
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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
−8.048092526059853195262617189355, −7.32159101793711739342272771899, −6.85382029568258713552968141223, −5.93534740724530577280092426663, −5.10907752777702365074607053766, −4.24489927926751819356759325327, −3.63296715940153736271108112339, −2.77654935595863057526663971769, −2.02172570993907659245615502199, −0.65153632609238689528768348354,
0.65153632609238689528768348354, 2.02172570993907659245615502199, 2.77654935595863057526663971769, 3.63296715940153736271108112339, 4.24489927926751819356759325327, 5.10907752777702365074607053766, 5.93534740724530577280092426663, 6.85382029568258713552968141223, 7.32159101793711739342272771899, 8.048092526059853195262617189355
