L(s) = 1 | − 2.45·2-s + 0.342·3-s + 4.04·4-s − 0.529·5-s − 0.841·6-s − 2.69·7-s − 5.01·8-s − 2.88·9-s + 1.30·10-s + 3.44·11-s + 1.38·12-s − 3.97·13-s + 6.61·14-s − 0.181·15-s + 4.24·16-s + 4.73·17-s + 7.08·18-s + 0.229·19-s − 2.14·20-s − 0.922·21-s − 8.46·22-s + 23-s − 1.71·24-s − 4.71·25-s + 9.77·26-s − 2.01·27-s − 10.8·28-s + ⋯ |
L(s) = 1 | − 1.73·2-s + 0.197·3-s + 2.02·4-s − 0.236·5-s − 0.343·6-s − 1.01·7-s − 1.77·8-s − 0.960·9-s + 0.411·10-s + 1.03·11-s + 0.399·12-s − 1.10·13-s + 1.76·14-s − 0.0468·15-s + 1.06·16-s + 1.14·17-s + 1.66·18-s + 0.0525·19-s − 0.478·20-s − 0.201·21-s − 1.80·22-s + 0.208·23-s − 0.350·24-s − 0.943·25-s + 1.91·26-s − 0.387·27-s − 2.05·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3922814340\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3922814340\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 23 | \( 1 - T \) |
| 349 | \( 1 + T \) |
good | 2 | \( 1 + 2.45T + 2T^{2} \) |
| 3 | \( 1 - 0.342T + 3T^{2} \) |
| 5 | \( 1 + 0.529T + 5T^{2} \) |
| 7 | \( 1 + 2.69T + 7T^{2} \) |
| 11 | \( 1 - 3.44T + 11T^{2} \) |
| 13 | \( 1 + 3.97T + 13T^{2} \) |
| 17 | \( 1 - 4.73T + 17T^{2} \) |
| 19 | \( 1 - 0.229T + 19T^{2} \) |
| 29 | \( 1 - 1.12T + 29T^{2} \) |
| 31 | \( 1 + 2.98T + 31T^{2} \) |
| 37 | \( 1 - 1.41T + 37T^{2} \) |
| 41 | \( 1 - 4.11T + 41T^{2} \) |
| 43 | \( 1 + 8.35T + 43T^{2} \) |
| 47 | \( 1 - 4.03T + 47T^{2} \) |
| 53 | \( 1 + 2.06T + 53T^{2} \) |
| 59 | \( 1 + 4.70T + 59T^{2} \) |
| 61 | \( 1 - 5.09T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 + 9.99T + 71T^{2} \) |
| 73 | \( 1 - 4.39T + 73T^{2} \) |
| 79 | \( 1 - 6.05T + 79T^{2} \) |
| 83 | \( 1 + 4.61T + 83T^{2} \) |
| 89 | \( 1 + 16.5T + 89T^{2} \) |
| 97 | \( 1 + 18.0T + 97T^{2} \) |
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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
−7.976771272727315156573628177654, −7.32450137121431611920119217217, −6.74404724308936113101465901557, −6.07279816017386242540574261966, −5.31502012948390277682513365948, −4.00279287990193480104009837737, −3.16953114565281399494728691288, −2.53188377570651701898527068234, −1.50525328140631615552925326488, −0.40300803416014330162943209212,
0.40300803416014330162943209212, 1.50525328140631615552925326488, 2.53188377570651701898527068234, 3.16953114565281399494728691288, 4.00279287990193480104009837737, 5.31502012948390277682513365948, 6.07279816017386242540574261966, 6.74404724308936113101465901557, 7.32450137121431611920119217217, 7.976771272727315156573628177654