L(s) = 1 | − 3·3-s + 2·7-s + 6·9-s − 5·13-s − 6·17-s − 6·19-s − 6·21-s − 23-s − 5·25-s − 9·27-s + 9·29-s − 3·31-s − 8·37-s + 15·39-s + 3·41-s + 8·43-s − 7·47-s − 3·49-s + 18·51-s − 2·53-s + 18·57-s − 4·59-s − 10·61-s + 12·63-s − 8·67-s + 3·69-s − 7·71-s + ⋯ |
L(s) = 1 | − 1.73·3-s + 0.755·7-s + 2·9-s − 1.38·13-s − 1.45·17-s − 1.37·19-s − 1.30·21-s − 0.208·23-s − 25-s − 1.73·27-s + 1.67·29-s − 0.538·31-s − 1.31·37-s + 2.40·39-s + 0.468·41-s + 1.21·43-s − 1.02·47-s − 3/7·49-s + 2.52·51-s − 0.274·53-s + 2.38·57-s − 0.520·59-s − 1.28·61-s + 1.51·63-s − 0.977·67-s + 0.361·69-s − 0.830·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−10.87685192818790698904347116728, −10.49478638439220047072956684207, −9.281471779609372196228294987777, −7.958342972325975140744296893865, −6.86475080211176163512795932999, −6.11946214096292349919315122416, −4.90940837161472249204141741393, −4.43460996824354305124974289687, −2.01709412933771627130438480353, 0,
2.01709412933771627130438480353, 4.43460996824354305124974289687, 4.90940837161472249204141741393, 6.11946214096292349919315122416, 6.86475080211176163512795932999, 7.958342972325975140744296893865, 9.281471779609372196228294987777, 10.49478638439220047072956684207, 10.87685192818790698904347116728