| L(s) = 1 | − 3-s + 1.41·7-s + 9-s + 5.65·11-s − 5.65·13-s + 2.24·17-s − 8.24·19-s − 1.41·21-s − 23-s − 27-s + 8.82·29-s − 1.17·31-s − 5.65·33-s + 3.17·37-s + 5.65·39-s − 2·41-s + 5.41·43-s + 10.4·47-s − 5·49-s − 2.24·51-s − 4.58·53-s + 8.24·57-s − 8.82·59-s + 3.17·61-s + 1.41·63-s + 7.75·67-s + 69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.534·7-s + 0.333·9-s + 1.70·11-s − 1.56·13-s + 0.543·17-s − 1.89·19-s − 0.308·21-s − 0.208·23-s − 0.192·27-s + 1.63·29-s − 0.210·31-s − 0.984·33-s + 0.521·37-s + 0.905·39-s − 0.312·41-s + 0.825·43-s + 1.52·47-s − 0.714·49-s − 0.314·51-s − 0.629·53-s + 1.09·57-s − 1.14·59-s + 0.406·61-s + 0.178·63-s + 0.947·67-s + 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.681326041\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.681326041\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 - 1.41T + 7T^{2} \) |
| 11 | \( 1 - 5.65T + 11T^{2} \) |
| 13 | \( 1 + 5.65T + 13T^{2} \) |
| 17 | \( 1 - 2.24T + 17T^{2} \) |
| 19 | \( 1 + 8.24T + 19T^{2} \) |
| 29 | \( 1 - 8.82T + 29T^{2} \) |
| 31 | \( 1 + 1.17T + 31T^{2} \) |
| 37 | \( 1 - 3.17T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 5.41T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 + 4.58T + 53T^{2} \) |
| 59 | \( 1 + 8.82T + 59T^{2} \) |
| 61 | \( 1 - 3.17T + 61T^{2} \) |
| 67 | \( 1 - 7.75T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 + 0.343T + 73T^{2} \) |
| 79 | \( 1 + 7.07T + 79T^{2} \) |
| 83 | \( 1 - 9.65T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 + 16.1T + 97T^{2} \) |
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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
−7.935516132176908969513267574725, −7.13493058425498781096394028896, −6.51401338591865452170252470312, −6.00904489141418492057349618822, −4.96169978412932048996701135616, −4.47804773846696218924252910521, −3.82111565204198566397713619367, −2.57854518882698008795910152319, −1.75039749263401345697982251342, −0.69559506746064996798344251107,
0.69559506746064996798344251107, 1.75039749263401345697982251342, 2.57854518882698008795910152319, 3.82111565204198566397713619367, 4.47804773846696218924252910521, 4.96169978412932048996701135616, 6.00904489141418492057349618822, 6.51401338591865452170252470312, 7.13493058425498781096394028896, 7.935516132176908969513267574725
