L(s) = 1 | − 1.24·2-s + 0.554·4-s − 1.80·7-s + 0.554·8-s + 2.24·14-s − 1.24·16-s + 0.445·17-s − 0.445·19-s + 25-s − 0.999·28-s − 1.24·29-s + 1.24·31-s + 0.999·32-s − 0.554·34-s − 1.80·37-s + 0.554·38-s + 1.80·41-s + 1.24·43-s + 0.445·47-s + 2.24·49-s − 1.24·50-s + 0.445·53-s − 1.00·56-s + 1.55·58-s − 1.55·62-s + 0.246·68-s + 1.24·73-s + ⋯ |
L(s) = 1 | − 1.24·2-s + 0.554·4-s − 1.80·7-s + 0.554·8-s + 2.24·14-s − 1.24·16-s + 0.445·17-s − 0.445·19-s + 25-s − 0.999·28-s − 1.24·29-s + 1.24·31-s + 0.999·32-s − 0.554·34-s − 1.80·37-s + 0.554·38-s + 1.80·41-s + 1.24·43-s + 0.445·47-s + 2.24·49-s − 1.24·50-s + 0.445·53-s − 1.00·56-s + 1.55·58-s − 1.55·62-s + 0.246·68-s + 1.24·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4253992387\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4253992387\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 223 | \( 1 - T \) |
good | 2 | \( 1 + 1.24T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 + 1.80T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - 0.445T + T^{2} \) |
| 19 | \( 1 + 0.445T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 1.24T + T^{2} \) |
| 31 | \( 1 - 1.24T + T^{2} \) |
| 37 | \( 1 + 1.80T + T^{2} \) |
| 41 | \( 1 - 1.80T + T^{2} \) |
| 43 | \( 1 - 1.24T + T^{2} \) |
| 47 | \( 1 - 0.445T + T^{2} \) |
| 53 | \( 1 - 0.445T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 1.24T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.24T + T^{2} \) |
| 89 | \( 1 - 1.80T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.140012431847698383929473320749, −8.962594211910987334964122187592, −7.84923393670752290726898216994, −7.12398489429404988847123597191, −6.47450052765339411035190481816, −5.58771830874828068140007296924, −4.31783596908809514715396970534, −3.35311409441622559457283609813, −2.31856857864155598588747751888, −0.75955837371950665842889698027,
0.75955837371950665842889698027, 2.31856857864155598588747751888, 3.35311409441622559457283609813, 4.31783596908809514715396970534, 5.58771830874828068140007296924, 6.47450052765339411035190481816, 7.12398489429404988847123597191, 7.84923393670752290726898216994, 8.962594211910987334964122187592, 9.140012431847698383929473320749