| L(s) = 1 | − 2.44·3-s − 1.44·5-s + 2.99·9-s + 1.55·11-s + 13-s + 3.55·15-s + 2.44·17-s − 5.44·19-s − 5.89·23-s − 2.89·25-s + 3.89·29-s + 1.44·31-s − 3.79·33-s − 3.55·37-s − 2.44·39-s − 1.10·41-s + 43-s − 4.34·45-s + 1.44·47-s − 5.99·51-s − 7.89·53-s − 2.24·55-s + 13.3·57-s − 14·59-s + 2·61-s − 1.44·65-s + 2.89·67-s + ⋯ |
| L(s) = 1 | − 1.41·3-s − 0.648·5-s + 0.999·9-s + 0.467·11-s + 0.277·13-s + 0.916·15-s + 0.594·17-s − 1.25·19-s − 1.23·23-s − 0.579·25-s + 0.724·29-s + 0.260·31-s − 0.661·33-s − 0.583·37-s − 0.392·39-s − 0.171·41-s + 0.152·43-s − 0.648·45-s + 0.211·47-s − 0.840·51-s − 1.08·53-s − 0.303·55-s + 1.76·57-s − 1.82·59-s + 0.256·61-s − 0.179·65-s + 0.354·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6659744622\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6659744622\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 2.44T + 3T^{2} \) |
| 5 | \( 1 + 1.44T + 5T^{2} \) |
| 11 | \( 1 - 1.55T + 11T^{2} \) |
| 17 | \( 1 - 2.44T + 17T^{2} \) |
| 19 | \( 1 + 5.44T + 19T^{2} \) |
| 23 | \( 1 + 5.89T + 23T^{2} \) |
| 29 | \( 1 - 3.89T + 29T^{2} \) |
| 31 | \( 1 - 1.44T + 31T^{2} \) |
| 37 | \( 1 + 3.55T + 37T^{2} \) |
| 41 | \( 1 + 1.10T + 41T^{2} \) |
| 43 | \( 1 - T + 43T^{2} \) |
| 47 | \( 1 - 1.44T + 47T^{2} \) |
| 53 | \( 1 + 7.89T + 53T^{2} \) |
| 59 | \( 1 + 14T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 2.89T + 67T^{2} \) |
| 71 | \( 1 - 7.55T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 + 11.8T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 - 6.55T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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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
−8.798772106184225285536196335381, −8.081538455527208371188214947018, −7.27922739571828070743884488593, −6.24018790693486286306002051263, −6.09266770036734299740367835407, −4.93761000827692571057122557010, −4.28186671022114994007035871071, −3.41373726259073871327280850401, −1.87333330443159406517556187339, −0.54986255372974326845782942662,
0.54986255372974326845782942662, 1.87333330443159406517556187339, 3.41373726259073871327280850401, 4.28186671022114994007035871071, 4.93761000827692571057122557010, 6.09266770036734299740367835407, 6.24018790693486286306002051263, 7.27922739571828070743884488593, 8.081538455527208371188214947018, 8.798772106184225285536196335381
