| L(s) = 1 | − 4·3-s − 4·7-s + 7·9-s + 16·13-s + 68·19-s + 16·21-s − 5·25-s + 8·27-s − 28·31-s + 112·37-s − 64·39-s − 16·43-s − 86·49-s − 272·57-s − 92·61-s − 28·63-s − 64·67-s − 212·73-s + 20·75-s + 44·79-s − 95·81-s − 64·91-s + 112·93-s + 244·97-s + 92·103-s + 172·109-s − 448·111-s + ⋯ |
| L(s) = 1 | − 4/3·3-s − 4/7·7-s + 7/9·9-s + 1.23·13-s + 3.57·19-s + 0.761·21-s − 1/5·25-s + 8/27·27-s − 0.903·31-s + 3.02·37-s − 1.64·39-s − 0.372·43-s − 1.75·49-s − 4.77·57-s − 1.50·61-s − 4/9·63-s − 0.955·67-s − 2.90·73-s + 4/15·75-s + 0.556·79-s − 1.17·81-s − 0.703·91-s + 1.20·93-s + 2.51·97-s + 0.893·103-s + 1.57·109-s − 4.03·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57600 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.248026798\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.248026798\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + 4 T + p^{2} T^{2} \) |
| 5 | $C_2$ | \( 1 + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 62 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 398 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 34 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 562 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 62 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 56 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 2642 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 2798 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 3998 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6782 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 46 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 32 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 7202 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 106 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 802 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 142 T + p^{2} T^{2} )( 1 + 142 T + p^{2} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 122 T + p^{2} T^{2} )^{2} \) |
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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
−11.84635163800132264612389136374, −11.61463529123126255680878263803, −11.35444133131044377004341116583, −10.91809737861071506096622982979, −10.13884717696738528414571348664, −9.910082970761950223351205460749, −9.267594117484508364717482258331, −9.005573745742292200058942456802, −8.025773303184323858916193764338, −7.55759314885657683149523693394, −7.19655001376276928028719173631, −6.33933711002365588561088727038, −5.94632777445677102714009961557, −5.68422698109558736905296363283, −4.92482270008296073106892430765, −4.39793582494968958516912021908, −3.25400737175870062306895408962, −3.14551828905361041951591584773, −1.45434820455017486105777129164, −0.72093817613293607114905671851,
0.72093817613293607114905671851, 1.45434820455017486105777129164, 3.14551828905361041951591584773, 3.25400737175870062306895408962, 4.39793582494968958516912021908, 4.92482270008296073106892430765, 5.68422698109558736905296363283, 5.94632777445677102714009961557, 6.33933711002365588561088727038, 7.19655001376276928028719173631, 7.55759314885657683149523693394, 8.025773303184323858916193764338, 9.005573745742292200058942456802, 9.267594117484508364717482258331, 9.910082970761950223351205460749, 10.13884717696738528414571348664, 10.91809737861071506096622982979, 11.35444133131044377004341116583, 11.61463529123126255680878263803, 11.84635163800132264612389136374
