L(s) = 1 | − 3.18·3-s − 2.71·5-s − 7-s + 7.14·9-s + 11-s + 13-s + 8.64·15-s − 7.22·17-s + 0.763·19-s + 3.18·21-s + 4.54·23-s + 2.37·25-s − 13.1·27-s + 7.28·29-s − 9.42·31-s − 3.18·33-s + 2.71·35-s + 0.730·37-s − 3.18·39-s − 6.84·41-s − 7.14·43-s − 19.3·45-s + 4.54·47-s + 49-s + 23.0·51-s − 8.30·53-s − 2.71·55-s + ⋯ |
L(s) = 1 | − 1.83·3-s − 1.21·5-s − 0.377·7-s + 2.38·9-s + 0.301·11-s + 0.277·13-s + 2.23·15-s − 1.75·17-s + 0.175·19-s + 0.694·21-s + 0.948·23-s + 0.475·25-s − 2.53·27-s + 1.35·29-s − 1.69·31-s − 0.554·33-s + 0.459·35-s + 0.120·37-s − 0.509·39-s − 1.06·41-s − 1.08·43-s − 2.89·45-s + 0.663·47-s + 0.142·49-s + 3.22·51-s − 1.14·53-s − 0.366·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3049315251\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3049315251\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 3 | \( 1 + 3.18T + 3T^{2} \) |
| 5 | \( 1 + 2.71T + 5T^{2} \) |
| 17 | \( 1 + 7.22T + 17T^{2} \) |
| 19 | \( 1 - 0.763T + 19T^{2} \) |
| 23 | \( 1 - 4.54T + 23T^{2} \) |
| 29 | \( 1 - 7.28T + 29T^{2} \) |
| 31 | \( 1 + 9.42T + 31T^{2} \) |
| 37 | \( 1 - 0.730T + 37T^{2} \) |
| 41 | \( 1 + 6.84T + 41T^{2} \) |
| 43 | \( 1 + 7.14T + 43T^{2} \) |
| 47 | \( 1 - 4.54T + 47T^{2} \) |
| 53 | \( 1 + 8.30T + 53T^{2} \) |
| 59 | \( 1 - 4.13T + 59T^{2} \) |
| 61 | \( 1 + 2.50T + 61T^{2} \) |
| 67 | \( 1 + 3.86T + 67T^{2} \) |
| 71 | \( 1 + 15.9T + 71T^{2} \) |
| 73 | \( 1 - 3.47T + 73T^{2} \) |
| 79 | \( 1 + 17.2T + 79T^{2} \) |
| 83 | \( 1 + 14.3T + 83T^{2} \) |
| 89 | \( 1 - 8.11T + 89T^{2} \) |
| 97 | \( 1 - 11.2T + 97T^{2} \) |
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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
−8.460512489797221080199297124692, −7.33717835152284767597782397625, −6.90416075825512444620652646162, −6.33460139447223410139330180906, −5.47441410976512040176653969787, −4.62390363918460134694528486717, −4.20169848174731508440088784099, −3.19581904684533324212790634949, −1.57526569700133273191591602418, −0.35626338327080995097521088751,
0.35626338327080995097521088751, 1.57526569700133273191591602418, 3.19581904684533324212790634949, 4.20169848174731508440088784099, 4.62390363918460134694528486717, 5.47441410976512040176653969787, 6.33460139447223410139330180906, 6.90416075825512444620652646162, 7.33717835152284767597782397625, 8.460512489797221080199297124692