| L(s) = 1 | + 1.94·3-s + 2.86·5-s − 4.75·7-s + 0.798·9-s + 4.96·11-s − 3.92·13-s + 5.58·15-s + 0.630·17-s + 2.17·19-s − 9.26·21-s + 7.57·23-s + 3.19·25-s − 4.29·27-s + 6.80·29-s − 3.23·31-s + 9.66·33-s − 13.6·35-s + 1.73·37-s − 7.64·39-s + 8.70·41-s + 12.7·43-s + 2.28·45-s − 2.02·47-s + 15.5·49-s + 1.22·51-s − 1.58·53-s + 14.2·55-s + ⋯ |
| L(s) = 1 | + 1.12·3-s + 1.28·5-s − 1.79·7-s + 0.266·9-s + 1.49·11-s − 1.08·13-s + 1.44·15-s + 0.153·17-s + 0.499·19-s − 2.02·21-s + 1.58·23-s + 0.639·25-s − 0.825·27-s + 1.26·29-s − 0.581·31-s + 1.68·33-s − 2.29·35-s + 0.285·37-s − 1.22·39-s + 1.35·41-s + 1.94·43-s + 0.340·45-s − 0.296·47-s + 2.22·49-s + 0.172·51-s − 0.217·53-s + 1.91·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.211369285\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.211369285\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 - 1.94T + 3T^{2} \) |
| 5 | \( 1 - 2.86T + 5T^{2} \) |
| 7 | \( 1 + 4.75T + 7T^{2} \) |
| 11 | \( 1 - 4.96T + 11T^{2} \) |
| 13 | \( 1 + 3.92T + 13T^{2} \) |
| 17 | \( 1 - 0.630T + 17T^{2} \) |
| 19 | \( 1 - 2.17T + 19T^{2} \) |
| 23 | \( 1 - 7.57T + 23T^{2} \) |
| 29 | \( 1 - 6.80T + 29T^{2} \) |
| 31 | \( 1 + 3.23T + 31T^{2} \) |
| 37 | \( 1 - 1.73T + 37T^{2} \) |
| 41 | \( 1 - 8.70T + 41T^{2} \) |
| 43 | \( 1 - 12.7T + 43T^{2} \) |
| 47 | \( 1 + 2.02T + 47T^{2} \) |
| 53 | \( 1 + 1.58T + 53T^{2} \) |
| 59 | \( 1 + 3.96T + 59T^{2} \) |
| 61 | \( 1 + 2.62T + 61T^{2} \) |
| 67 | \( 1 + 4.95T + 67T^{2} \) |
| 71 | \( 1 - 5.00T + 71T^{2} \) |
| 73 | \( 1 + 10.6T + 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 - 5.48T + 83T^{2} \) |
| 89 | \( 1 + 2.32T + 89T^{2} \) |
| 97 | \( 1 - 16.6T + 97T^{2} \) |
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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
−8.925898122687189544488117339733, −7.58874412064023294068588285097, −6.97116006023521799779049559025, −6.25860295647562208186458254665, −5.72752463685428724348799693572, −4.55333102635624038188076912271, −3.51381798025201396399644944074, −2.89251729391359858253908360184, −2.29072487869861669853992367464, −0.993377380747892827034713957262,
0.993377380747892827034713957262, 2.29072487869861669853992367464, 2.89251729391359858253908360184, 3.51381798025201396399644944074, 4.55333102635624038188076912271, 5.72752463685428724348799693572, 6.25860295647562208186458254665, 6.97116006023521799779049559025, 7.58874412064023294068588285097, 8.925898122687189544488117339733
