| L(s) = 1 | + 3-s + 2.31·5-s + 4.58·7-s + 9-s + 3.21·11-s − 5.90·13-s + 2.31·15-s + 6.31·17-s − 7.08·19-s + 4.58·21-s − 1.12·23-s + 0.350·25-s + 27-s − 0.0956·29-s + 4.81·31-s + 3.21·33-s + 10.6·35-s + 3.60·37-s − 5.90·39-s + 8.59·41-s − 8.44·43-s + 2.31·45-s + 6.42·47-s + 14.0·49-s + 6.31·51-s + 2.56·53-s + 7.43·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.03·5-s + 1.73·7-s + 0.333·9-s + 0.969·11-s − 1.63·13-s + 0.597·15-s + 1.53·17-s − 1.62·19-s + 1.00·21-s − 0.233·23-s + 0.0700·25-s + 0.192·27-s − 0.0177·29-s + 0.865·31-s + 0.559·33-s + 1.79·35-s + 0.593·37-s − 0.946·39-s + 1.34·41-s − 1.28·43-s + 0.344·45-s + 0.937·47-s + 2.00·49-s + 0.883·51-s + 0.351·53-s + 1.00·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.756798815\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.756798815\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 - 2.31T + 5T^{2} \) |
| 7 | \( 1 - 4.58T + 7T^{2} \) |
| 11 | \( 1 - 3.21T + 11T^{2} \) |
| 13 | \( 1 + 5.90T + 13T^{2} \) |
| 17 | \( 1 - 6.31T + 17T^{2} \) |
| 19 | \( 1 + 7.08T + 19T^{2} \) |
| 23 | \( 1 + 1.12T + 23T^{2} \) |
| 29 | \( 1 + 0.0956T + 29T^{2} \) |
| 31 | \( 1 - 4.81T + 31T^{2} \) |
| 37 | \( 1 - 3.60T + 37T^{2} \) |
| 41 | \( 1 - 8.59T + 41T^{2} \) |
| 43 | \( 1 + 8.44T + 43T^{2} \) |
| 47 | \( 1 - 6.42T + 47T^{2} \) |
| 53 | \( 1 - 2.56T + 53T^{2} \) |
| 59 | \( 1 - 6.13T + 59T^{2} \) |
| 61 | \( 1 + 5.66T + 61T^{2} \) |
| 67 | \( 1 + 9.24T + 67T^{2} \) |
| 71 | \( 1 + 7.95T + 71T^{2} \) |
| 73 | \( 1 - 14.9T + 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 + 3.20T + 83T^{2} \) |
| 89 | \( 1 + 16.0T + 89T^{2} \) |
| 97 | \( 1 - 0.476T + 97T^{2} \) |
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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
−8.352950683091227130244266366883, −7.85134142330747854946897961989, −7.12812590564455591180727575293, −6.18161237605486147527178935773, −5.40895877314317540235508295240, −4.65363707230646831700723110504, −4.02470775141396009402871524832, −2.62067657964600691705809484343, −2.01699940121795650077591486045, −1.21133205348591590088771980689,
1.21133205348591590088771980689, 2.01699940121795650077591486045, 2.62067657964600691705809484343, 4.02470775141396009402871524832, 4.65363707230646831700723110504, 5.40895877314317540235508295240, 6.18161237605486147527178935773, 7.12812590564455591180727575293, 7.85134142330747854946897961989, 8.352950683091227130244266366883
