L(s) = 1 | − 2·2-s + 4·4-s + 10·5-s + 7·7-s − 8·8-s − 20·10-s − 8·11-s − 50·13-s − 14·14-s + 16·16-s − 114·17-s + 19·19-s + 40·20-s + 16·22-s + 148·23-s − 25·25-s + 100·26-s + 28·28-s + 30·29-s + 304·31-s − 32·32-s + 228·34-s + 70·35-s − 274·37-s − 38·38-s − 80·40-s + 202·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.377·7-s − 0.353·8-s − 0.632·10-s − 0.219·11-s − 1.06·13-s − 0.267·14-s + 1/4·16-s − 1.62·17-s + 0.229·19-s + 0.447·20-s + 0.155·22-s + 1.34·23-s − 1/5·25-s + 0.754·26-s + 0.188·28-s + 0.192·29-s + 1.76·31-s − 0.176·32-s + 1.15·34-s + 0.338·35-s − 1.21·37-s − 0.162·38-s − 0.316·40-s + 0.769·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + p T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - p T \) |
| 19 | \( 1 - p T \) |
good | 5 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 11 | \( 1 + 8 T + p^{3} T^{2} \) |
| 13 | \( 1 + 50 T + p^{3} T^{2} \) |
| 17 | \( 1 + 114 T + p^{3} T^{2} \) |
| 23 | \( 1 - 148 T + p^{3} T^{2} \) |
| 29 | \( 1 - 30 T + p^{3} T^{2} \) |
| 31 | \( 1 - 304 T + p^{3} T^{2} \) |
| 37 | \( 1 + 274 T + p^{3} T^{2} \) |
| 41 | \( 1 - 202 T + p^{3} T^{2} \) |
| 43 | \( 1 + 116 T + p^{3} T^{2} \) |
| 47 | \( 1 - 324 T + p^{3} T^{2} \) |
| 53 | \( 1 - 550 T + p^{3} T^{2} \) |
| 59 | \( 1 + 628 T + p^{3} T^{2} \) |
| 61 | \( 1 + 58 T + p^{3} T^{2} \) |
| 67 | \( 1 + 756 T + p^{3} T^{2} \) |
| 71 | \( 1 - 216 T + p^{3} T^{2} \) |
| 73 | \( 1 + 278 T + p^{3} T^{2} \) |
| 79 | \( 1 + 952 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1184 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1542 T + p^{3} T^{2} \) |
| 97 | \( 1 + 870 T + p^{3} T^{2} \) |
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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
−8.425398285888150257616978913904, −7.40399186802543261518996593762, −6.85268757751299862731943155820, −6.00581360934126391753699133931, −5.10862375676951993458250424912, −4.38245245971149070948713368972, −2.82657944644151904778975881842, −2.26067686156543308061415249168, −1.23445907955584138301360236049, 0,
1.23445907955584138301360236049, 2.26067686156543308061415249168, 2.82657944644151904778975881842, 4.38245245971149070948713368972, 5.10862375676951993458250424912, 6.00581360934126391753699133931, 6.85268757751299862731943155820, 7.40399186802543261518996593762, 8.425398285888150257616978913904