L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 5·7-s + 8-s + 9-s − 11-s + 12-s − 5·13-s + 5·14-s + 16-s + 4·17-s + 18-s − 2·19-s + 5·21-s − 22-s − 4·23-s + 24-s − 5·26-s + 27-s + 5·28-s + 6·29-s − 31-s + 32-s − 33-s + 4·34-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.88·7-s + 0.353·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s − 1.38·13-s + 1.33·14-s + 1/4·16-s + 0.970·17-s + 0.235·18-s − 0.458·19-s + 1.09·21-s − 0.213·22-s − 0.834·23-s + 0.204·24-s − 0.980·26-s + 0.192·27-s + 0.944·28-s + 1.11·29-s − 0.179·31-s + 0.176·32-s − 0.174·33-s + 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.814312909\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.814312909\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 31 | \( 1 + T \) |
good | 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 9 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.107181278313114094064727835157, −7.60303713826352618995471873716, −7.12426339042036056642412756520, −5.83802237631748727094569608874, −5.27719803880453546388322663458, −4.48227964836929338115814256970, −4.05601323745177517663726272260, −2.68261923148758089141122888628, −2.23183263827833193034387148837, −1.15368464176892312015444190252,
1.15368464176892312015444190252, 2.23183263827833193034387148837, 2.68261923148758089141122888628, 4.05601323745177517663726272260, 4.48227964836929338115814256970, 5.27719803880453546388322663458, 5.83802237631748727094569608874, 7.12426339042036056642412756520, 7.60303713826352618995471873716, 8.107181278313114094064727835157