L(s) = 1 | + 3-s − 2·4-s − 5-s + 9-s − 2·12-s − 13-s − 15-s + 4·16-s + 6·17-s + 5·19-s + 2·20-s + 6·23-s + 25-s + 27-s − 6·29-s + 5·31-s − 2·36-s − 7·37-s − 39-s + 12·41-s − 43-s − 45-s + 6·47-s + 4·48-s + 6·51-s + 2·52-s + 5·57-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 4-s − 0.447·5-s + 1/3·9-s − 0.577·12-s − 0.277·13-s − 0.258·15-s + 16-s + 1.45·17-s + 1.14·19-s + 0.447·20-s + 1.25·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s + 0.898·31-s − 1/3·36-s − 1.15·37-s − 0.160·39-s + 1.87·41-s − 0.152·43-s − 0.149·45-s + 0.875·47-s + 0.577·48-s + 0.840·51-s + 0.277·52-s + 0.662·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.429863258\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.429863258\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 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
−10.10865226012453578347249877301, −9.457335487461132856891366279938, −8.746165196706298306495639523635, −7.79365059547869888962371823747, −7.27697356506803510184292872836, −5.69549232653133514693898609833, −4.88443961818724254544969070577, −3.79488993654363240578018568649, −2.99421930943913221166038508817, −1.04599175459547455544259640973,
1.04599175459547455544259640973, 2.99421930943913221166038508817, 3.79488993654363240578018568649, 4.88443961818724254544969070577, 5.69549232653133514693898609833, 7.27697356506803510184292872836, 7.79365059547869888962371823747, 8.746165196706298306495639523635, 9.457335487461132856891366279938, 10.10865226012453578347249877301