L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s + 5·11-s − 12-s + 16-s − 4·17-s − 18-s − 8·19-s − 5·22-s + 4·23-s + 24-s − 27-s − 5·29-s − 3·31-s − 32-s − 5·33-s + 4·34-s + 36-s + 4·37-s + 8·38-s − 2·43-s + 5·44-s − 4·46-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 1.50·11-s − 0.288·12-s + 1/4·16-s − 0.970·17-s − 0.235·18-s − 1.83·19-s − 1.06·22-s + 0.834·23-s + 0.204·24-s − 0.192·27-s − 0.928·29-s − 0.538·31-s − 0.176·32-s − 0.870·33-s + 0.685·34-s + 1/6·36-s + 0.657·37-s + 1.29·38-s − 0.304·43-s + 0.753·44-s − 0.589·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9892654951\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9892654951\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−7.996812276476719793015482831467, −6.89768837137028799084453361173, −6.75439238624848250504641117037, −6.05488968082577638003451534230, −5.17315100556416877500146073327, −4.22112753860125750391990261354, −3.72409296785306934538231080597, −2.38494125894939686418316767282, −1.66261522591419602494227629899, −0.58208683539537247822388290160,
0.58208683539537247822388290160, 1.66261522591419602494227629899, 2.38494125894939686418316767282, 3.72409296785306934538231080597, 4.22112753860125750391990261354, 5.17315100556416877500146073327, 6.05488968082577638003451534230, 6.75439238624848250504641117037, 6.89768837137028799084453361173, 7.996812276476719793015482831467