L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s + 5·7-s − 8-s + 9-s − 10-s + 12-s + 5·13-s − 5·14-s + 15-s + 16-s − 3·17-s − 18-s − 19-s + 20-s + 5·21-s − 6·23-s − 24-s + 25-s − 5·26-s + 27-s + 5·28-s + 9·29-s − 30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 1.88·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s + 1.38·13-s − 1.33·14-s + 0.258·15-s + 1/4·16-s − 0.727·17-s − 0.235·18-s − 0.229·19-s + 0.223·20-s + 1.09·21-s − 1.25·23-s − 0.204·24-s + 1/5·25-s − 0.980·26-s + 0.192·27-s + 0.944·28-s + 1.67·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.563579768\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.563579768\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 7 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−8.669412667447228725678463767663, −8.011444476926069450264432367847, −7.36534906352138831598701636951, −6.36147055815058117261348836021, −5.68828340130227770333571928249, −4.61178783539340463690511981667, −3.97767015973847203075182772516, −2.63642778949948222087926353177, −1.84663775138695948501597486318, −1.13258632403368171248471985680,
1.13258632403368171248471985680, 1.84663775138695948501597486318, 2.63642778949948222087926353177, 3.97767015973847203075182772516, 4.61178783539340463690511981667, 5.68828340130227770333571928249, 6.36147055815058117261348836021, 7.36534906352138831598701636951, 8.011444476926069450264432367847, 8.669412667447228725678463767663