| L(s) = 1 | − 2.41·2-s + 3.82·4-s + 2.82·5-s + 3·7-s − 4.41·8-s − 6.82·10-s + 0.828·11-s + 6.65·13-s − 7.24·14-s + 2.99·16-s − 3·19-s + 10.8·20-s − 1.99·22-s + 2.82·23-s + 3.00·25-s − 16.0·26-s + 11.4·28-s + 3.17·29-s − 4.65·31-s + 1.58·32-s + 8.48·35-s + 6.65·37-s + 7.24·38-s − 12.4·40-s − 8.82·41-s + 3·43-s + 3.17·44-s + ⋯ |
| L(s) = 1 | − 1.70·2-s + 1.91·4-s + 1.26·5-s + 1.13·7-s − 1.56·8-s − 2.15·10-s + 0.249·11-s + 1.84·13-s − 1.93·14-s + 0.749·16-s − 0.688·19-s + 2.42·20-s − 0.426·22-s + 0.589·23-s + 0.600·25-s − 3.15·26-s + 2.17·28-s + 0.588·29-s − 0.836·31-s + 0.280·32-s + 1.43·35-s + 1.09·37-s + 1.17·38-s − 1.97·40-s − 1.37·41-s + 0.457·43-s + 0.478·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.387888518\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.387888518\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 5 | \( 1 - 2.82T + 5T^{2} \) |
| 7 | \( 1 - 3T + 7T^{2} \) |
| 11 | \( 1 - 0.828T + 11T^{2} \) |
| 13 | \( 1 - 6.65T + 13T^{2} \) |
| 19 | \( 1 + 3T + 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 - 3.17T + 29T^{2} \) |
| 31 | \( 1 + 4.65T + 31T^{2} \) |
| 37 | \( 1 - 6.65T + 37T^{2} \) |
| 41 | \( 1 + 8.82T + 41T^{2} \) |
| 43 | \( 1 - 3T + 43T^{2} \) |
| 47 | \( 1 - 9.65T + 47T^{2} \) |
| 53 | \( 1 - 1.65T + 53T^{2} \) |
| 59 | \( 1 - 8.82T + 59T^{2} \) |
| 61 | \( 1 + 4.65T + 61T^{2} \) |
| 67 | \( 1 - T + 67T^{2} \) |
| 71 | \( 1 - 5.17T + 71T^{2} \) |
| 73 | \( 1 + 7.65T + 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 + 1.17T + 83T^{2} \) |
| 89 | \( 1 + 5.31T + 89T^{2} \) |
| 97 | \( 1 + 10.6T + 97T^{2} \) |
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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
−8.766306113751004986411677935148, −8.471001141926281018605666393340, −7.62574616970448908786372576114, −6.66501547023393278891792819801, −6.10677028762625800774189172064, −5.24304870316674471175434210571, −3.99852586754188614984565566127, −2.54358060923041425958698476751, −1.66885470530098542883582767622, −1.08722723542063213747532443087,
1.08722723542063213747532443087, 1.66885470530098542883582767622, 2.54358060923041425958698476751, 3.99852586754188614984565566127, 5.24304870316674471175434210571, 6.10677028762625800774189172064, 6.66501547023393278891792819801, 7.62574616970448908786372576114, 8.471001141926281018605666393340, 8.766306113751004986411677935148
