| L(s) = 1 | + 2·2-s + 2·3-s + 2·4-s + 4·6-s + 3·9-s + 10·11-s + 4·12-s − 4·16-s − 14·17-s + 6·18-s − 12·19-s + 20·22-s + 25-s + 4·27-s − 8·32-s + 20·33-s − 28·34-s + 6·36-s − 24·38-s + 18·41-s − 16·43-s + 20·44-s − 8·48-s − 13·49-s + 2·50-s − 28·51-s + 8·54-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 4-s + 1.63·6-s + 9-s + 3.01·11-s + 1.15·12-s − 16-s − 3.39·17-s + 1.41·18-s − 2.75·19-s + 4.26·22-s + 1/5·25-s + 0.769·27-s − 1.41·32-s + 3.48·33-s − 4.80·34-s + 36-s − 3.89·38-s + 2.81·41-s − 2.43·43-s + 3.01·44-s − 1.15·48-s − 1.85·49-s + 0.282·50-s − 3.92·51-s + 1.08·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2433600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2433600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.99490652189327518276262034176, −6.92169485416648038433791572415, −6.48261927781574407124221142201, −6.28215256608024799923334344773, −6.09015330024649050428370832049, −4.96208675506281559301750823763, −4.52648172003467775210616179348, −4.30670154917300743784663424717, −3.93328920262448493706980455054, −3.83380892295724030655167419744, −2.88146428193362253481394458287, −2.53020079052262607160250914109, −1.82495831022597068255399157880, −1.64397492516826605581886054973, 0,
1.64397492516826605581886054973, 1.82495831022597068255399157880, 2.53020079052262607160250914109, 2.88146428193362253481394458287, 3.83380892295724030655167419744, 3.93328920262448493706980455054, 4.30670154917300743784663424717, 4.52648172003467775210616179348, 4.96208675506281559301750823763, 6.09015330024649050428370832049, 6.28215256608024799923334344773, 6.48261927781574407124221142201, 6.92169485416648038433791572415, 6.99490652189327518276262034176
