| L(s) = 1 | + 3-s + 2.56·5-s − 7-s + 9-s + 2·11-s − 13-s + 2.56·15-s − 3.12·17-s + 6.56·19-s − 21-s + 7.68·23-s + 1.56·25-s + 27-s + 0.561·29-s + 2.56·31-s + 2·33-s − 2.56·35-s − 7.12·37-s − 39-s − 1.12·41-s + 5.43·43-s + 2.56·45-s − 5.68·47-s + 49-s − 3.12·51-s + 4.56·53-s + 5.12·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.14·5-s − 0.377·7-s + 0.333·9-s + 0.603·11-s − 0.277·13-s + 0.661·15-s − 0.757·17-s + 1.50·19-s − 0.218·21-s + 1.60·23-s + 0.312·25-s + 0.192·27-s + 0.104·29-s + 0.460·31-s + 0.348·33-s − 0.432·35-s − 1.17·37-s − 0.160·39-s − 0.175·41-s + 0.829·43-s + 0.381·45-s − 0.829·47-s + 0.142·49-s − 0.437·51-s + 0.626·53-s + 0.690·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.192432307\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.192432307\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 2.56T + 5T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 17 | \( 1 + 3.12T + 17T^{2} \) |
| 19 | \( 1 - 6.56T + 19T^{2} \) |
| 23 | \( 1 - 7.68T + 23T^{2} \) |
| 29 | \( 1 - 0.561T + 29T^{2} \) |
| 31 | \( 1 - 2.56T + 31T^{2} \) |
| 37 | \( 1 + 7.12T + 37T^{2} \) |
| 41 | \( 1 + 1.12T + 41T^{2} \) |
| 43 | \( 1 - 5.43T + 43T^{2} \) |
| 47 | \( 1 + 5.68T + 47T^{2} \) |
| 53 | \( 1 - 4.56T + 53T^{2} \) |
| 59 | \( 1 - 3.12T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 - 14.8T + 73T^{2} \) |
| 79 | \( 1 + 8.80T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 - 1.43T + 89T^{2} \) |
| 97 | \( 1 + 3.43T + 97T^{2} \) |
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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
−8.592031143377707751291020430237, −7.53123188168722198939407970556, −6.89818681012205918023028989733, −6.28203039892872926885278307966, −5.37480572561691179871554656684, −4.74352431333538609399444049399, −3.59691775495615444920228810348, −2.88340205073481354091870599228, −2.01154362202121127135375199172, −1.03569791237203781219273122564,
1.03569791237203781219273122564, 2.01154362202121127135375199172, 2.88340205073481354091870599228, 3.59691775495615444920228810348, 4.74352431333538609399444049399, 5.37480572561691179871554656684, 6.28203039892872926885278307966, 6.89818681012205918023028989733, 7.53123188168722198939407970556, 8.592031143377707751291020430237
