L(s) = 1 | + 3-s + 5-s + 7-s + 9-s + 11-s + 4.91·13-s + 15-s + 1.15·17-s + 8.20·19-s + 21-s − 3.90·23-s + 25-s + 27-s − 4.06·29-s + 4.91·31-s + 33-s + 35-s + 9.96·37-s + 4.91·39-s − 7.96·41-s − 9.42·43-s + 45-s − 0.163·47-s + 49-s + 1.15·51-s − 6.97·53-s + 55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.377·7-s + 0.333·9-s + 0.301·11-s + 1.36·13-s + 0.258·15-s + 0.279·17-s + 1.88·19-s + 0.218·21-s − 0.813·23-s + 0.200·25-s + 0.192·27-s − 0.754·29-s + 0.882·31-s + 0.174·33-s + 0.169·35-s + 1.63·37-s + 0.786·39-s − 1.24·41-s − 1.43·43-s + 0.149·45-s − 0.0238·47-s + 0.142·49-s + 0.161·51-s − 0.958·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.330408373\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.330408373\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 - 4.91T + 13T^{2} \) |
| 17 | \( 1 - 1.15T + 17T^{2} \) |
| 19 | \( 1 - 8.20T + 19T^{2} \) |
| 23 | \( 1 + 3.90T + 23T^{2} \) |
| 29 | \( 1 + 4.06T + 29T^{2} \) |
| 31 | \( 1 - 4.91T + 31T^{2} \) |
| 37 | \( 1 - 9.96T + 37T^{2} \) |
| 41 | \( 1 + 7.96T + 41T^{2} \) |
| 43 | \( 1 + 9.42T + 43T^{2} \) |
| 47 | \( 1 + 0.163T + 47T^{2} \) |
| 53 | \( 1 + 6.97T + 53T^{2} \) |
| 59 | \( 1 - 1.75T + 59T^{2} \) |
| 61 | \( 1 + 6.51T + 61T^{2} \) |
| 67 | \( 1 - 1.22T + 67T^{2} \) |
| 71 | \( 1 + 2.14T + 71T^{2} \) |
| 73 | \( 1 - 9.35T + 73T^{2} \) |
| 79 | \( 1 + 17.0T + 79T^{2} \) |
| 83 | \( 1 - 15.7T + 83T^{2} \) |
| 89 | \( 1 - 1.31T + 89T^{2} \) |
| 97 | \( 1 + 18.9T + 97T^{2} \) |
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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
−8.202625092906318166002064219732, −7.82810252722934969610644868529, −6.87108651016305165138742831411, −6.12208198802195493067403613112, −5.43736354937898388196580036899, −4.54970929223658819051098078092, −3.60945024143560655088295057740, −3.02123100517547948901364421485, −1.81059527237161515764699229762, −1.09970786067538245656463253941,
1.09970786067538245656463253941, 1.81059527237161515764699229762, 3.02123100517547948901364421485, 3.60945024143560655088295057740, 4.54970929223658819051098078092, 5.43736354937898388196580036899, 6.12208198802195493067403613112, 6.87108651016305165138742831411, 7.82810252722934969610644868529, 8.202625092906318166002064219732