| L(s) = 1 | + 1.95·3-s + 2.85·5-s − 2.57·7-s + 0.823·9-s + 1.95·11-s + 2.85·13-s + 5.59·15-s + 7.03·17-s + 4.97·19-s − 5.03·21-s − 8.60·23-s + 3.17·25-s − 4.25·27-s − 29-s + 4.53·31-s + 3.82·33-s − 7.36·35-s − 7.71·37-s + 5.59·39-s + 6.68·41-s − 3.19·43-s + 2.35·45-s + 3.29·47-s − 0.365·49-s + 13.7·51-s + 3.21·53-s + 5.59·55-s + ⋯ |
| L(s) = 1 | + 1.12·3-s + 1.27·5-s − 0.973·7-s + 0.274·9-s + 0.589·11-s + 0.793·13-s + 1.44·15-s + 1.70·17-s + 1.14·19-s − 1.09·21-s − 1.79·23-s + 0.635·25-s − 0.819·27-s − 0.185·29-s + 0.813·31-s + 0.665·33-s − 1.24·35-s − 1.26·37-s + 0.895·39-s + 1.04·41-s − 0.487·43-s + 0.350·45-s + 0.479·47-s − 0.0521·49-s + 1.92·51-s + 0.441·53-s + 0.753·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.183131540\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.183131540\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 1.95T + 3T^{2} \) |
| 5 | \( 1 - 2.85T + 5T^{2} \) |
| 7 | \( 1 + 2.57T + 7T^{2} \) |
| 11 | \( 1 - 1.95T + 11T^{2} \) |
| 13 | \( 1 - 2.85T + 13T^{2} \) |
| 17 | \( 1 - 7.03T + 17T^{2} \) |
| 19 | \( 1 - 4.97T + 19T^{2} \) |
| 23 | \( 1 + 8.60T + 23T^{2} \) |
| 31 | \( 1 - 4.53T + 31T^{2} \) |
| 37 | \( 1 + 7.71T + 37T^{2} \) |
| 41 | \( 1 - 6.68T + 41T^{2} \) |
| 43 | \( 1 + 3.19T + 43T^{2} \) |
| 47 | \( 1 - 3.29T + 47T^{2} \) |
| 53 | \( 1 - 3.21T + 53T^{2} \) |
| 59 | \( 1 - 13.7T + 59T^{2} \) |
| 61 | \( 1 - 7.03T + 61T^{2} \) |
| 67 | \( 1 + 7.82T + 67T^{2} \) |
| 71 | \( 1 - 6.03T + 71T^{2} \) |
| 73 | \( 1 - 1.64T + 73T^{2} \) |
| 79 | \( 1 + 0.620T + 79T^{2} \) |
| 83 | \( 1 + 16.4T + 83T^{2} \) |
| 89 | \( 1 - 7.39T + 89T^{2} \) |
| 97 | \( 1 + 4.75T + 97T^{2} \) |
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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
−9.362368223793549524483179201161, −8.560589711346051211330007708233, −7.83362528199757429720904511556, −6.84926735836285048645682571774, −5.90846482396670879871159634416, −5.55355812020295728244016481789, −3.85365132420693348143172614856, −3.30273388875026636026718691620, −2.35613715984118305307818103047, −1.29088727631478881414330741131,
1.29088727631478881414330741131, 2.35613715984118305307818103047, 3.30273388875026636026718691620, 3.85365132420693348143172614856, 5.55355812020295728244016481789, 5.90846482396670879871159634416, 6.84926735836285048645682571774, 7.83362528199757429720904511556, 8.560589711346051211330007708233, 9.362368223793549524483179201161
