L(s) = 1 | − 7-s − 3·9-s − 13-s + 6·17-s − 4·19-s − 23-s − 5·25-s − 2·29-s − 10·31-s + 4·37-s + 10·41-s − 4·43-s + 6·47-s + 49-s − 2·53-s + 2·59-s + 10·61-s + 3·63-s − 16·67-s + 10·71-s − 2·73-s − 8·79-s + 9·81-s + 16·83-s − 8·89-s + 91-s − 4·97-s + ⋯ |
L(s) = 1 | − 0.377·7-s − 9-s − 0.277·13-s + 1.45·17-s − 0.917·19-s − 0.208·23-s − 25-s − 0.371·29-s − 1.79·31-s + 0.657·37-s + 1.56·41-s − 0.609·43-s + 0.875·47-s + 1/7·49-s − 0.274·53-s + 0.260·59-s + 1.28·61-s + 0.377·63-s − 1.95·67-s + 1.18·71-s − 0.234·73-s − 0.900·79-s + 81-s + 1.75·83-s − 0.847·89-s + 0.104·91-s − 0.406·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8372 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8372 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.239976735\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.239976735\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 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
−7.72571681840542982291064439482, −7.28247620406321655134688675437, −6.20209145774496462984655513146, −5.79955865654546976534178849542, −5.19792678884844518458920461858, −4.10937685537398688705916782420, −3.51504155545576214970796905728, −2.67319577356266528395990922059, −1.86392002529938651726262629798, −0.52540714455550545417067338156,
0.52540714455550545417067338156, 1.86392002529938651726262629798, 2.67319577356266528395990922059, 3.51504155545576214970796905728, 4.10937685537398688705916782420, 5.19792678884844518458920461858, 5.79955865654546976534178849542, 6.20209145774496462984655513146, 7.28247620406321655134688675437, 7.72571681840542982291064439482