L(s) = 1 | + 2·3-s − 4·4-s − 2·7-s − 3·9-s + 6·11-s − 8·12-s + 12·16-s − 4·21-s − 10·25-s − 14·27-s + 8·28-s + 12·33-s + 12·36-s + 37-s − 18·41-s − 24·44-s + 6·47-s + 24·48-s − 11·49-s − 6·53-s + 6·63-s − 32·64-s − 8·67-s − 30·71-s + 22·73-s − 20·75-s − 12·77-s + ⋯ |
L(s) = 1 | + 1.15·3-s − 2·4-s − 0.755·7-s − 9-s + 1.80·11-s − 2.30·12-s + 3·16-s − 0.872·21-s − 2·25-s − 2.69·27-s + 1.51·28-s + 2.08·33-s + 2·36-s + 0.164·37-s − 2.81·41-s − 3.61·44-s + 0.875·47-s + 3.46·48-s − 1.57·49-s − 0.824·53-s + 0.755·63-s − 4·64-s − 0.977·67-s − 3.56·71-s + 2.57·73-s − 2.30·75-s − 1.36·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50653 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50653 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 37 | $C_1$ | \( 1 - T \) |
good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−9.761720804850964159547106328126, −9.033994536698006403547335776331, −9.032345851694468357832029856913, −8.514386701951993332004045317832, −7.966542131008976901646671712207, −7.59911177067371416247669368567, −6.45112619828749871048432130507, −6.08767175428500734181159353176, −5.44973416215471530225317007593, −4.72629665865280345114046599990, −3.94489242393102636413792410970, −3.50910294340479626549928321873, −3.18533882088337592342139702542, −1.75988621528230231849959555830, 0,
1.75988621528230231849959555830, 3.18533882088337592342139702542, 3.50910294340479626549928321873, 3.94489242393102636413792410970, 4.72629665865280345114046599990, 5.44973416215471530225317007593, 6.08767175428500734181159353176, 6.45112619828749871048432130507, 7.59911177067371416247669368567, 7.966542131008976901646671712207, 8.514386701951993332004045317832, 9.032345851694468357832029856913, 9.033994536698006403547335776331, 9.761720804850964159547106328126