| L(s) = 1 | − 3-s − 1.38·5-s − 7-s + 9-s + 5.47·11-s − 2.38·13-s + 1.38·15-s − 17-s + 3·19-s + 21-s + 23-s − 3.09·25-s − 27-s − 7.47·29-s + 3.76·31-s − 5.47·33-s + 1.38·35-s + 1.47·37-s + 2.38·39-s − 4.70·41-s − 8.09·43-s − 1.38·45-s − 1.70·47-s + 49-s + 51-s − 3.38·53-s − 7.56·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.618·5-s − 0.377·7-s + 0.333·9-s + 1.64·11-s − 0.660·13-s + 0.356·15-s − 0.242·17-s + 0.688·19-s + 0.218·21-s + 0.208·23-s − 0.618·25-s − 0.192·27-s − 1.38·29-s + 0.676·31-s − 0.952·33-s + 0.233·35-s + 0.242·37-s + 0.381·39-s − 0.735·41-s − 1.23·43-s − 0.206·45-s − 0.249·47-s + 0.142·49-s + 0.140·51-s − 0.464·53-s − 1.01·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 1.38T + 5T^{2} \) |
| 11 | \( 1 - 5.47T + 11T^{2} \) |
| 13 | \( 1 + 2.38T + 13T^{2} \) |
| 17 | \( 1 + T + 17T^{2} \) |
| 19 | \( 1 - 3T + 19T^{2} \) |
| 29 | \( 1 + 7.47T + 29T^{2} \) |
| 31 | \( 1 - 3.76T + 31T^{2} \) |
| 37 | \( 1 - 1.47T + 37T^{2} \) |
| 41 | \( 1 + 4.70T + 41T^{2} \) |
| 43 | \( 1 + 8.09T + 43T^{2} \) |
| 47 | \( 1 + 1.70T + 47T^{2} \) |
| 53 | \( 1 + 3.38T + 53T^{2} \) |
| 59 | \( 1 - 6.14T + 59T^{2} \) |
| 61 | \( 1 - 13.7T + 61T^{2} \) |
| 67 | \( 1 + 4.14T + 67T^{2} \) |
| 71 | \( 1 - 3.90T + 71T^{2} \) |
| 73 | \( 1 - 2.70T + 73T^{2} \) |
| 79 | \( 1 + 0.527T + 79T^{2} \) |
| 83 | \( 1 - 3T + 83T^{2} \) |
| 89 | \( 1 - 3.14T + 89T^{2} \) |
| 97 | \( 1 - 5T + 97T^{2} \) |
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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
−7.33583996745382366769884987302, −6.83503088498859371027033549492, −6.24309040386397337219214827844, −5.41322544637923895601106400271, −4.67620866390625816571551744568, −3.85666330852308709926961893959, −3.40994181849326957461612661570, −2.13055224009501831231306687664, −1.12415943668422468051484352214, 0,
1.12415943668422468051484352214, 2.13055224009501831231306687664, 3.40994181849326957461612661570, 3.85666330852308709926961893959, 4.67620866390625816571551744568, 5.41322544637923895601106400271, 6.24309040386397337219214827844, 6.83503088498859371027033549492, 7.33583996745382366769884987302
