| L(s) = 1 | − 0.408·3-s + 4.38·5-s − 2.83·9-s + 4.19·11-s + 2.35·13-s − 1.79·15-s + 3.64·17-s − 19-s + 5.23·23-s + 14.2·25-s + 2.38·27-s + 3.13·29-s + 1.82·31-s − 1.71·33-s + 4.56·37-s − 0.962·39-s + 0.293·41-s − 7.18·43-s − 12.4·45-s + 0.248·47-s − 1.48·51-s + 1.96·53-s + 18.3·55-s + 0.408·57-s − 14.0·59-s − 5.55·61-s + 10.3·65-s + ⋯ |
| L(s) = 1 | − 0.235·3-s + 1.96·5-s − 0.944·9-s + 1.26·11-s + 0.653·13-s − 0.462·15-s + 0.884·17-s − 0.229·19-s + 1.09·23-s + 2.84·25-s + 0.458·27-s + 0.582·29-s + 0.327·31-s − 0.298·33-s + 0.749·37-s − 0.154·39-s + 0.0458·41-s − 1.09·43-s − 1.85·45-s + 0.0361·47-s − 0.208·51-s + 0.269·53-s + 2.48·55-s + 0.0541·57-s − 1.83·59-s − 0.711·61-s + 1.28·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.341586516\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.341586516\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 0.408T + 3T^{2} \) |
| 5 | \( 1 - 4.38T + 5T^{2} \) |
| 11 | \( 1 - 4.19T + 11T^{2} \) |
| 13 | \( 1 - 2.35T + 13T^{2} \) |
| 17 | \( 1 - 3.64T + 17T^{2} \) |
| 23 | \( 1 - 5.23T + 23T^{2} \) |
| 29 | \( 1 - 3.13T + 29T^{2} \) |
| 31 | \( 1 - 1.82T + 31T^{2} \) |
| 37 | \( 1 - 4.56T + 37T^{2} \) |
| 41 | \( 1 - 0.293T + 41T^{2} \) |
| 43 | \( 1 + 7.18T + 43T^{2} \) |
| 47 | \( 1 - 0.248T + 47T^{2} \) |
| 53 | \( 1 - 1.96T + 53T^{2} \) |
| 59 | \( 1 + 14.0T + 59T^{2} \) |
| 61 | \( 1 + 5.55T + 61T^{2} \) |
| 67 | \( 1 + 12.9T + 67T^{2} \) |
| 71 | \( 1 - 4.44T + 71T^{2} \) |
| 73 | \( 1 + 6.61T + 73T^{2} \) |
| 79 | \( 1 + 0.470T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 - 17.3T + 89T^{2} \) |
| 97 | \( 1 + 3.71T + 97T^{2} \) |
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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
−8.018220709777445606350976784243, −6.83987571489105956809526871572, −6.35314840543137090995775755086, −5.91706473170944714875070131284, −5.27303067529135406940719201336, −4.53366439226127118548698426728, −3.28978325674559723715823308757, −2.72990987328793857940245017412, −1.62616826317935088448418500000, −1.03910097453208432868006020021,
1.03910097453208432868006020021, 1.62616826317935088448418500000, 2.72990987328793857940245017412, 3.28978325674559723715823308757, 4.53366439226127118548698426728, 5.27303067529135406940719201336, 5.91706473170944714875070131284, 6.35314840543137090995775755086, 6.83987571489105956809526871572, 8.018220709777445606350976784243
