| L(s) = 1 | + 0.704·3-s + 3.89·5-s + 7-s − 2.50·9-s + 1.15·11-s + 4.05·13-s + 2.74·15-s − 5.09·17-s + 4.35·19-s + 0.704·21-s + 2.70·23-s + 10.1·25-s − 3.87·27-s − 2.74·29-s − 1.89·31-s + 0.811·33-s + 3.89·35-s − 2.57·37-s + 2.85·39-s + 41-s + 7.09·43-s − 9.74·45-s − 0.312·47-s + 49-s − 3.58·51-s − 3.89·53-s + 4.48·55-s + ⋯ |
| L(s) = 1 | + 0.406·3-s + 1.74·5-s + 0.377·7-s − 0.834·9-s + 0.347·11-s + 1.12·13-s + 0.707·15-s − 1.23·17-s + 0.998·19-s + 0.153·21-s + 0.563·23-s + 2.03·25-s − 0.745·27-s − 0.509·29-s − 0.340·31-s + 0.141·33-s + 0.658·35-s − 0.423·37-s + 0.457·39-s + 0.156·41-s + 1.08·43-s − 1.45·45-s − 0.0455·47-s + 0.142·49-s − 0.502·51-s − 0.534·53-s + 0.604·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.623483622\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.623483622\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 - 0.704T + 3T^{2} \) |
| 5 | \( 1 - 3.89T + 5T^{2} \) |
| 11 | \( 1 - 1.15T + 11T^{2} \) |
| 13 | \( 1 - 4.05T + 13T^{2} \) |
| 17 | \( 1 + 5.09T + 17T^{2} \) |
| 19 | \( 1 - 4.35T + 19T^{2} \) |
| 23 | \( 1 - 2.70T + 23T^{2} \) |
| 29 | \( 1 + 2.74T + 29T^{2} \) |
| 31 | \( 1 + 1.89T + 31T^{2} \) |
| 37 | \( 1 + 2.57T + 37T^{2} \) |
| 43 | \( 1 - 7.09T + 43T^{2} \) |
| 47 | \( 1 + 0.312T + 47T^{2} \) |
| 53 | \( 1 + 3.89T + 53T^{2} \) |
| 59 | \( 1 - 0.811T + 59T^{2} \) |
| 61 | \( 1 + 15.1T + 61T^{2} \) |
| 67 | \( 1 + 5.41T + 67T^{2} \) |
| 71 | \( 1 + 2.88T + 71T^{2} \) |
| 73 | \( 1 + 6.93T + 73T^{2} \) |
| 79 | \( 1 - 13.8T + 79T^{2} \) |
| 83 | \( 1 - 3.11T + 83T^{2} \) |
| 89 | \( 1 - 4.61T + 89T^{2} \) |
| 97 | \( 1 + 0.112T + 97T^{2} \) |
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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
−9.466025889686882809316629523351, −9.131891404911141318312813155936, −8.454275610315360972524395434176, −7.26709555685744027636706627296, −6.20784204063297934339615870645, −5.75140777318495686234317202878, −4.77022951869634601733212887300, −3.38275846346359902645233666285, −2.36610499158064648167325941948, −1.40841568927235903542340738869,
1.40841568927235903542340738869, 2.36610499158064648167325941948, 3.38275846346359902645233666285, 4.77022951869634601733212887300, 5.75140777318495686234317202878, 6.20784204063297934339615870645, 7.26709555685744027636706627296, 8.454275610315360972524395434176, 9.131891404911141318312813155936, 9.466025889686882809316629523351
