| L(s) = 1 | + 3.56·5-s − 3.56·7-s + 2·11-s − 13-s − 3.56·17-s − 6·19-s + 7.68·25-s − 8.24·29-s + 1.12·31-s − 12.6·35-s + 2.68·37-s + 1.12·41-s − 11.8·43-s − 10.6·47-s + 5.68·49-s + 13.1·53-s + 7.12·55-s − 6·59-s − 11.3·61-s − 3.56·65-s − 6·67-s + 10.6·71-s + 10·73-s − 7.12·77-s − 12·79-s + 7.36·83-s − 12.6·85-s + ⋯ |
| L(s) = 1 | + 1.59·5-s − 1.34·7-s + 0.603·11-s − 0.277·13-s − 0.863·17-s − 1.37·19-s + 1.53·25-s − 1.53·29-s + 0.201·31-s − 2.14·35-s + 0.441·37-s + 0.175·41-s − 1.80·43-s − 1.55·47-s + 0.812·49-s + 1.80·53-s + 0.960·55-s − 0.781·59-s − 1.45·61-s − 0.441·65-s − 0.733·67-s + 1.26·71-s + 1.17·73-s − 0.811·77-s − 1.35·79-s + 0.808·83-s − 1.37·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 3.56T + 5T^{2} \) |
| 7 | \( 1 + 3.56T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 17 | \( 1 + 3.56T + 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 8.24T + 29T^{2} \) |
| 31 | \( 1 - 1.12T + 31T^{2} \) |
| 37 | \( 1 - 2.68T + 37T^{2} \) |
| 41 | \( 1 - 1.12T + 41T^{2} \) |
| 43 | \( 1 + 11.8T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 + 6T + 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 - 7.36T + 83T^{2} \) |
| 89 | \( 1 + 8.24T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−8.359247470924614125970790125691, −7.09046374063523696582329303791, −6.45394742968289184001431464816, −6.16318255530587707759055538034, −5.28505144718888428024789098250, −4.30570908196938274634522827852, −3.33471575680941272079501992305, −2.38466120869572987838380453888, −1.67036395773389906988686940996, 0,
1.67036395773389906988686940996, 2.38466120869572987838380453888, 3.33471575680941272079501992305, 4.30570908196938274634522827852, 5.28505144718888428024789098250, 6.16318255530587707759055538034, 6.45394742968289184001431464816, 7.09046374063523696582329303791, 8.359247470924614125970790125691
