| L(s) = 1 | − 1.35·3-s + 5-s + 0.606·7-s − 1.17·9-s − 0.513·11-s − 4.05·13-s − 1.35·15-s + 1.21·17-s − 3.41·19-s − 0.819·21-s + 5.56·23-s + 25-s + 5.64·27-s − 0.352·29-s − 3.19·31-s + 0.693·33-s + 0.606·35-s + 2.97·37-s + 5.47·39-s + 9.88·41-s − 5.95·43-s − 1.17·45-s + 6.73·47-s − 6.63·49-s − 1.64·51-s + 13.8·53-s − 0.513·55-s + ⋯ |
| L(s) = 1 | − 0.780·3-s + 0.447·5-s + 0.229·7-s − 0.391·9-s − 0.154·11-s − 1.12·13-s − 0.348·15-s + 0.294·17-s − 0.783·19-s − 0.178·21-s + 1.16·23-s + 0.200·25-s + 1.08·27-s − 0.0654·29-s − 0.574·31-s + 0.120·33-s + 0.102·35-s + 0.489·37-s + 0.877·39-s + 1.54·41-s − 0.907·43-s − 0.175·45-s + 0.982·47-s − 0.947·49-s − 0.230·51-s + 1.90·53-s − 0.0692·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 3 | \( 1 + 1.35T + 3T^{2} \) |
| 7 | \( 1 - 0.606T + 7T^{2} \) |
| 11 | \( 1 + 0.513T + 11T^{2} \) |
| 13 | \( 1 + 4.05T + 13T^{2} \) |
| 17 | \( 1 - 1.21T + 17T^{2} \) |
| 19 | \( 1 + 3.41T + 19T^{2} \) |
| 23 | \( 1 - 5.56T + 23T^{2} \) |
| 29 | \( 1 + 0.352T + 29T^{2} \) |
| 31 | \( 1 + 3.19T + 31T^{2} \) |
| 37 | \( 1 - 2.97T + 37T^{2} \) |
| 41 | \( 1 - 9.88T + 41T^{2} \) |
| 43 | \( 1 + 5.95T + 43T^{2} \) |
| 47 | \( 1 - 6.73T + 47T^{2} \) |
| 53 | \( 1 - 13.8T + 53T^{2} \) |
| 59 | \( 1 + 9.57T + 59T^{2} \) |
| 61 | \( 1 - 2.75T + 61T^{2} \) |
| 67 | \( 1 + 0.461T + 67T^{2} \) |
| 71 | \( 1 - 16.2T + 71T^{2} \) |
| 73 | \( 1 + 14.9T + 73T^{2} \) |
| 79 | \( 1 + 7.62T + 79T^{2} \) |
| 83 | \( 1 - 13.6T + 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 + 16.8T + 97T^{2} \) |
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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
−7.35878873375084159656753332167, −6.75929240139574069771132789097, −6.00173108778376162013680735256, −5.38653442272964347449598091050, −4.89587252611337329057138597715, −4.08034406452362645416645092897, −2.89952090889738261263342793508, −2.31463312494112335188610897024, −1.12012499479711526182043725994, 0,
1.12012499479711526182043725994, 2.31463312494112335188610897024, 2.89952090889738261263342793508, 4.08034406452362645416645092897, 4.89587252611337329057138597715, 5.38653442272964347449598091050, 6.00173108778376162013680735256, 6.75929240139574069771132789097, 7.35878873375084159656753332167
