L(s) = 1 | − 2-s + 4-s + 5-s + 7-s − 8-s − 10-s + 11-s − 4·13-s − 14-s + 16-s − 4·19-s + 20-s − 22-s + 25-s + 4·26-s + 28-s + 6·29-s − 10·31-s − 32-s + 35-s + 2·37-s + 4·38-s − 40-s + 12·41-s − 4·43-s + 44-s − 6·47-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s − 0.353·8-s − 0.316·10-s + 0.301·11-s − 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.917·19-s + 0.223·20-s − 0.213·22-s + 1/5·25-s + 0.784·26-s + 0.188·28-s + 1.11·29-s − 1.79·31-s − 0.176·32-s + 0.169·35-s + 0.328·37-s + 0.648·38-s − 0.158·40-s + 1.87·41-s − 0.609·43-s + 0.150·44-s − 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−7.53720325615662210345950957175, −7.10730850495843010027290080247, −6.28273015267067949583013594897, −5.61684761980580521068539892725, −4.77726893102678202067266622317, −4.02732123477456736800703918177, −2.84991005125484048890506098875, −2.17803474014677912313912607860, −1.30231631584590740742244911776, 0,
1.30231631584590740742244911776, 2.17803474014677912313912607860, 2.84991005125484048890506098875, 4.02732123477456736800703918177, 4.77726893102678202067266622317, 5.61684761980580521068539892725, 6.28273015267067949583013594897, 7.10730850495843010027290080247, 7.53720325615662210345950957175