| L(s) = 1 | + 1.41·3-s − 2.44·7-s − 0.999·9-s + 3.46·11-s + 4.89·17-s − 3.46·19-s − 3.46·21-s − 2.44·23-s − 5.65·27-s − 4·31-s + 4.89·33-s − 8.48·37-s + 4.24·43-s + 7.34·47-s − 1.00·49-s + 6.92·51-s − 5.65·53-s − 4.89·57-s − 10.3·59-s − 3.46·61-s + 2.44·63-s − 4.24·67-s − 3.46·69-s − 12·71-s + 4.89·73-s − 8.48·77-s − 4·79-s + ⋯ |
| L(s) = 1 | + 0.816·3-s − 0.925·7-s − 0.333·9-s + 1.04·11-s + 1.18·17-s − 0.794·19-s − 0.755·21-s − 0.510·23-s − 1.08·27-s − 0.718·31-s + 0.852·33-s − 1.39·37-s + 0.646·43-s + 1.07·47-s − 0.142·49-s + 0.970·51-s − 0.777·53-s − 0.648·57-s − 1.35·59-s − 0.443·61-s + 0.308·63-s − 0.518·67-s − 0.417·69-s − 1.42·71-s + 0.573·73-s − 0.966·77-s − 0.450·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 - 1.41T + 3T^{2} \) |
| 7 | \( 1 + 2.44T + 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 4.89T + 17T^{2} \) |
| 19 | \( 1 + 3.46T + 19T^{2} \) |
| 23 | \( 1 + 2.44T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 8.48T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 4.24T + 43T^{2} \) |
| 47 | \( 1 - 7.34T + 47T^{2} \) |
| 53 | \( 1 + 5.65T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 + 3.46T + 61T^{2} \) |
| 67 | \( 1 + 4.24T + 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 4.89T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 9.89T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 4.89T + 97T^{2} \) |
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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
−7.70529786306474780427275903570, −7.06386113853892077842035925396, −6.16712617429737877471979552790, −5.80197538562869542651259440784, −4.65630692974678314939297179735, −3.64191608388784036272786074950, −3.38857621263984253441279364423, −2.41056262494820274165743846940, −1.45306303919754013875753306283, 0,
1.45306303919754013875753306283, 2.41056262494820274165743846940, 3.38857621263984253441279364423, 3.64191608388784036272786074950, 4.65630692974678314939297179735, 5.80197538562869542651259440784, 6.16712617429737877471979552790, 7.06386113853892077842035925396, 7.70529786306474780427275903570
