| L(s) = 1 | + 1.40·3-s − 0.422·5-s + 4.72·7-s − 1.02·9-s + 3.71·11-s + 13-s − 0.593·15-s + 2.39·17-s + 8.15·19-s + 6.64·21-s − 7.87·23-s − 4.82·25-s − 5.65·27-s − 5.87·29-s + 0.528·31-s + 5.22·33-s − 1.99·35-s + 1.20·37-s + 1.40·39-s + 9.03·41-s + 2.19·43-s + 0.431·45-s − 11.1·47-s + 15.2·49-s + 3.36·51-s − 5.03·53-s − 1.56·55-s + ⋯ |
| L(s) = 1 | + 0.811·3-s − 0.188·5-s + 1.78·7-s − 0.340·9-s + 1.12·11-s + 0.277·13-s − 0.153·15-s + 0.579·17-s + 1.87·19-s + 1.44·21-s − 1.64·23-s − 0.964·25-s − 1.08·27-s − 1.09·29-s + 0.0948·31-s + 0.909·33-s − 0.336·35-s + 0.198·37-s + 0.225·39-s + 1.41·41-s + 0.334·43-s + 0.0643·45-s − 1.62·47-s + 2.18·49-s + 0.470·51-s − 0.691·53-s − 0.211·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.821561118\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.821561118\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 1.40T + 3T^{2} \) |
| 5 | \( 1 + 0.422T + 5T^{2} \) |
| 7 | \( 1 - 4.72T + 7T^{2} \) |
| 11 | \( 1 - 3.71T + 11T^{2} \) |
| 17 | \( 1 - 2.39T + 17T^{2} \) |
| 19 | \( 1 - 8.15T + 19T^{2} \) |
| 23 | \( 1 + 7.87T + 23T^{2} \) |
| 29 | \( 1 + 5.87T + 29T^{2} \) |
| 31 | \( 1 - 0.528T + 31T^{2} \) |
| 37 | \( 1 - 1.20T + 37T^{2} \) |
| 41 | \( 1 - 9.03T + 41T^{2} \) |
| 43 | \( 1 - 2.19T + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 + 5.03T + 53T^{2} \) |
| 59 | \( 1 - 3.71T + 59T^{2} \) |
| 61 | \( 1 - 14.6T + 61T^{2} \) |
| 67 | \( 1 - 6.47T + 67T^{2} \) |
| 71 | \( 1 - 9.34T + 71T^{2} \) |
| 73 | \( 1 + 16.5T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 - 6.08T + 83T^{2} \) |
| 89 | \( 1 - 8.63T + 89T^{2} \) |
| 97 | \( 1 + 0.755T + 97T^{2} \) |
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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
−9.337398609024212450208136899572, −8.375256040791567878355469970800, −7.910017554533815109109194421049, −7.37793141380188615302330126579, −5.95945389958151148515670579785, −5.30437914844158417248896119696, −4.13887956939734380800317341547, −3.52326307638047871594345500713, −2.17837302482165484361027577066, −1.29658404404318300490955748804,
1.29658404404318300490955748804, 2.17837302482165484361027577066, 3.52326307638047871594345500713, 4.13887956939734380800317341547, 5.30437914844158417248896119696, 5.95945389958151148515670579785, 7.37793141380188615302330126579, 7.910017554533815109109194421049, 8.375256040791567878355469970800, 9.337398609024212450208136899572
