| L(s) = 1 | − 2.41·3-s + 2.82·9-s + 1.82·11-s − 6.41·13-s − 3.58·17-s − 7.65·19-s − 3.41·23-s + 0.414·27-s − 4.65·29-s + 7.41·31-s − 4.41·33-s + 0.585·37-s + 15.4·39-s + 3.41·41-s − 0.343·43-s − 10.8·47-s + 8.65·51-s + 12.2·53-s + 18.4·57-s + 0.585·59-s + 10.8·61-s − 3.07·67-s + 8.24·69-s − 10.4·71-s − 10.8·73-s − 15.1·79-s − 9.48·81-s + ⋯ |
| L(s) = 1 | − 1.39·3-s + 0.942·9-s + 0.551·11-s − 1.77·13-s − 0.869·17-s − 1.75·19-s − 0.711·23-s + 0.0797·27-s − 0.864·29-s + 1.33·31-s − 0.768·33-s + 0.0963·37-s + 2.47·39-s + 0.533·41-s − 0.0523·43-s − 1.58·47-s + 1.21·51-s + 1.68·53-s + 2.44·57-s + 0.0762·59-s + 1.38·61-s − 0.375·67-s + 0.992·69-s − 1.24·71-s − 1.26·73-s − 1.70·79-s − 1.05·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4484616935\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4484616935\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2.41T + 3T^{2} \) |
| 11 | \( 1 - 1.82T + 11T^{2} \) |
| 13 | \( 1 + 6.41T + 13T^{2} \) |
| 17 | \( 1 + 3.58T + 17T^{2} \) |
| 19 | \( 1 + 7.65T + 19T^{2} \) |
| 23 | \( 1 + 3.41T + 23T^{2} \) |
| 29 | \( 1 + 4.65T + 29T^{2} \) |
| 31 | \( 1 - 7.41T + 31T^{2} \) |
| 37 | \( 1 - 0.585T + 37T^{2} \) |
| 41 | \( 1 - 3.41T + 41T^{2} \) |
| 43 | \( 1 + 0.343T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 - 0.585T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 + 3.07T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 + 15.1T + 79T^{2} \) |
| 83 | \( 1 - 8T + 83T^{2} \) |
| 89 | \( 1 + 16.9T + 89T^{2} \) |
| 97 | \( 1 - 9.72T + 97T^{2} \) |
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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
−8.266933201226701603892161721164, −7.25850841463244811661236018799, −6.71650169398138830427316184516, −6.11505677467749651311572160514, −5.40208378944222566037328558096, −4.50522039866236258392907338659, −4.22913823865939928500260710265, −2.69559801808095764890736257734, −1.84240914351221378005378898685, −0.37845424944339418819621958065,
0.37845424944339418819621958065, 1.84240914351221378005378898685, 2.69559801808095764890736257734, 4.22913823865939928500260710265, 4.50522039866236258392907338659, 5.40208378944222566037328558096, 6.11505677467749651311572160514, 6.71650169398138830427316184516, 7.25850841463244811661236018799, 8.266933201226701603892161721164
