L(s) = 1 | + 3-s − 3.95·5-s + 1.63·7-s + 9-s + 4.82·11-s − 5.59·13-s − 3.95·15-s − 0.828·17-s − 2.82·19-s + 1.63·21-s + 7.91·23-s + 10.6·25-s + 27-s + 7.23·29-s − 1.63·31-s + 4.82·33-s − 6.48·35-s + 2.31·37-s − 5.59·39-s + 3.17·41-s + 4.48·43-s − 3.95·45-s + 7.91·47-s − 4.31·49-s − 0.828·51-s + 0.678·53-s − 19.1·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.76·5-s + 0.619·7-s + 0.333·9-s + 1.45·11-s − 1.55·13-s − 1.02·15-s − 0.200·17-s − 0.648·19-s + 0.357·21-s + 1.65·23-s + 2.13·25-s + 0.192·27-s + 1.34·29-s − 0.294·31-s + 0.840·33-s − 1.09·35-s + 0.381·37-s − 0.896·39-s + 0.495·41-s + 0.683·43-s − 0.589·45-s + 1.15·47-s − 0.616·49-s − 0.116·51-s + 0.0932·53-s − 2.57·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.629373107\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.629373107\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
good | 5 | \( 1 + 3.95T + 5T^{2} \) |
| 7 | \( 1 - 1.63T + 7T^{2} \) |
| 11 | \( 1 - 4.82T + 11T^{2} \) |
| 13 | \( 1 + 5.59T + 13T^{2} \) |
| 17 | \( 1 + 0.828T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 - 7.91T + 23T^{2} \) |
| 29 | \( 1 - 7.23T + 29T^{2} \) |
| 31 | \( 1 + 1.63T + 31T^{2} \) |
| 37 | \( 1 - 2.31T + 37T^{2} \) |
| 41 | \( 1 - 3.17T + 41T^{2} \) |
| 43 | \( 1 - 4.48T + 43T^{2} \) |
| 47 | \( 1 - 7.91T + 47T^{2} \) |
| 53 | \( 1 - 0.678T + 53T^{2} \) |
| 59 | \( 1 - 9.65T + 59T^{2} \) |
| 61 | \( 1 - 2.31T + 61T^{2} \) |
| 67 | \( 1 + 13.6T + 67T^{2} \) |
| 71 | \( 1 + 3.27T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 - 1.63T + 79T^{2} \) |
| 83 | \( 1 - 8.82T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
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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
−9.070607675552095609242114780060, −8.754613380528066396220468603874, −7.74833601986276248342257014573, −7.29553132270502361943727247639, −6.54329110643883996842409073755, −4.84740150890825885802562223034, −4.40809179445598129659432166342, −3.53839627562226145909601721397, −2.49457424504648060187593979008, −0.901332249465280244979342675905,
0.901332249465280244979342675905, 2.49457424504648060187593979008, 3.53839627562226145909601721397, 4.40809179445598129659432166342, 4.84740150890825885802562223034, 6.54329110643883996842409073755, 7.29553132270502361943727247639, 7.74833601986276248342257014573, 8.754613380528066396220468603874, 9.070607675552095609242114780060