L(s) = 1 | + 2-s − 3-s + 4-s + 3.18·5-s − 6-s + 8-s + 9-s + 3.18·10-s − 1.67·11-s − 12-s − 2.27·13-s − 3.18·15-s + 16-s − 0.367·17-s + 18-s + 3.41·19-s + 3.18·20-s − 1.67·22-s − 23-s − 24-s + 5.13·25-s − 2.27·26-s − 27-s + 3.44·29-s − 3.18·30-s + 3.41·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.42·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s + 1.00·10-s − 0.504·11-s − 0.288·12-s − 0.630·13-s − 0.822·15-s + 0.250·16-s − 0.0892·17-s + 0.235·18-s + 0.783·19-s + 0.711·20-s − 0.356·22-s − 0.208·23-s − 0.204·24-s + 1.02·25-s − 0.445·26-s − 0.192·27-s + 0.639·29-s − 0.581·30-s + 0.613·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.531968286\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.531968286\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 - 3.18T + 5T^{2} \) |
| 11 | \( 1 + 1.67T + 11T^{2} \) |
| 13 | \( 1 + 2.27T + 13T^{2} \) |
| 17 | \( 1 + 0.367T + 17T^{2} \) |
| 19 | \( 1 - 3.41T + 19T^{2} \) |
| 29 | \( 1 - 3.44T + 29T^{2} \) |
| 31 | \( 1 - 3.41T + 31T^{2} \) |
| 37 | \( 1 + 0.0953T + 37T^{2} \) |
| 41 | \( 1 - 0.681T + 41T^{2} \) |
| 43 | \( 1 + 7.94T + 43T^{2} \) |
| 47 | \( 1 - 1.64T + 47T^{2} \) |
| 53 | \( 1 - 9.69T + 53T^{2} \) |
| 59 | \( 1 - 3.15T + 59T^{2} \) |
| 61 | \( 1 - 13.9T + 61T^{2} \) |
| 67 | \( 1 - 5.19T + 67T^{2} \) |
| 71 | \( 1 - 1.67T + 71T^{2} \) |
| 73 | \( 1 - 11.1T + 73T^{2} \) |
| 79 | \( 1 + 4.32T + 79T^{2} \) |
| 83 | \( 1 + 4.74T + 83T^{2} \) |
| 89 | \( 1 - 11.3T + 89T^{2} \) |
| 97 | \( 1 - 8.29T + 97T^{2} \) |
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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
−7.80551220153631486367849652365, −6.95397782068828674717002152539, −6.47150315904340753831361994563, −5.68440953513841601105471791722, −5.24686856499377897241529361914, −4.69069084795955345197958000281, −3.61688126153777712584343268895, −2.60402264845893627692030673159, −2.03365831166524500584554889129, −0.903814998778094796770615592642,
0.903814998778094796770615592642, 2.03365831166524500584554889129, 2.60402264845893627692030673159, 3.61688126153777712584343268895, 4.69069084795955345197958000281, 5.24686856499377897241529361914, 5.68440953513841601105471791722, 6.47150315904340753831361994563, 6.95397782068828674717002152539, 7.80551220153631486367849652365