| L(s) = 1 | + 2.94·3-s − 2.68·5-s + 0.723·7-s + 5.65·9-s − 11-s − 0.951·13-s − 7.90·15-s − 0.367·17-s − 2.91·19-s + 2.12·21-s + 0.0834·23-s + 2.22·25-s + 7.82·27-s − 8.77·29-s − 6.01·31-s − 2.94·33-s − 1.94·35-s + 10.1·37-s − 2.79·39-s − 5.88·41-s − 8.70·43-s − 15.2·45-s − 8.78·47-s − 6.47·49-s − 1.08·51-s + 5.17·53-s + 2.68·55-s + ⋯ |
| L(s) = 1 | + 1.69·3-s − 1.20·5-s + 0.273·7-s + 1.88·9-s − 0.301·11-s − 0.263·13-s − 2.04·15-s − 0.0890·17-s − 0.669·19-s + 0.464·21-s + 0.0173·23-s + 0.445·25-s + 1.50·27-s − 1.62·29-s − 1.08·31-s − 0.512·33-s − 0.328·35-s + 1.66·37-s − 0.448·39-s − 0.918·41-s − 1.32·43-s − 2.26·45-s − 1.28·47-s − 0.925·49-s − 0.151·51-s + 0.710·53-s + 0.362·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 137 | \( 1 - T \) |
| good | 3 | \( 1 - 2.94T + 3T^{2} \) |
| 5 | \( 1 + 2.68T + 5T^{2} \) |
| 7 | \( 1 - 0.723T + 7T^{2} \) |
| 13 | \( 1 + 0.951T + 13T^{2} \) |
| 17 | \( 1 + 0.367T + 17T^{2} \) |
| 19 | \( 1 + 2.91T + 19T^{2} \) |
| 23 | \( 1 - 0.0834T + 23T^{2} \) |
| 29 | \( 1 + 8.77T + 29T^{2} \) |
| 31 | \( 1 + 6.01T + 31T^{2} \) |
| 37 | \( 1 - 10.1T + 37T^{2} \) |
| 41 | \( 1 + 5.88T + 41T^{2} \) |
| 43 | \( 1 + 8.70T + 43T^{2} \) |
| 47 | \( 1 + 8.78T + 47T^{2} \) |
| 53 | \( 1 - 5.17T + 53T^{2} \) |
| 59 | \( 1 + 3.81T + 59T^{2} \) |
| 61 | \( 1 - 14.5T + 61T^{2} \) |
| 67 | \( 1 - 5.57T + 67T^{2} \) |
| 71 | \( 1 - 2.81T + 71T^{2} \) |
| 73 | \( 1 + 2.36T + 73T^{2} \) |
| 79 | \( 1 + 2.46T + 79T^{2} \) |
| 83 | \( 1 + 15.5T + 83T^{2} \) |
| 89 | \( 1 - 16.0T + 89T^{2} \) |
| 97 | \( 1 + 16.2T + 97T^{2} \) |
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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
−7.957300919565037372979306657403, −7.31327128462745538151543908401, −6.67046940035440466287447868421, −5.39619691780353833016086251882, −4.50500316604860394694147537832, −3.82321038360263064897873147220, −3.35390131156055273859251943966, −2.41178807884288986625277776750, −1.64748847412385359832229678882, 0,
1.64748847412385359832229678882, 2.41178807884288986625277776750, 3.35390131156055273859251943966, 3.82321038360263064897873147220, 4.50500316604860394694147537832, 5.39619691780353833016086251882, 6.67046940035440466287447868421, 7.31327128462745538151543908401, 7.957300919565037372979306657403
