L(s) = 1 | + (0.674 + 0.737i)2-s + (0.0296 + 0.999i)3-s + (−0.0887 + 0.996i)4-s + (0.263 + 0.964i)5-s + (−0.717 + 0.696i)6-s + (−0.430 − 0.902i)7-s + (−0.794 + 0.606i)8-s + (−0.998 + 0.0592i)9-s + (−0.533 + 0.845i)10-s + (0.829 − 0.558i)11-s + (−0.998 − 0.0592i)12-s + (0.582 − 0.812i)13-s + (0.375 − 0.926i)14-s + (−0.956 + 0.292i)15-s + (−0.984 − 0.176i)16-s + (0.482 − 0.875i)17-s + ⋯ |
L(s) = 1 | + (0.674 + 0.737i)2-s + (0.0296 + 0.999i)3-s + (−0.0887 + 0.996i)4-s + (0.263 + 0.964i)5-s + (−0.717 + 0.696i)6-s + (−0.430 − 0.902i)7-s + (−0.794 + 0.606i)8-s + (−0.998 + 0.0592i)9-s + (−0.533 + 0.845i)10-s + (0.829 − 0.558i)11-s + (−0.998 − 0.0592i)12-s + (0.582 − 0.812i)13-s + (0.375 − 0.926i)14-s + (−0.956 + 0.292i)15-s + (−0.984 − 0.176i)16-s + (0.482 − 0.875i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.714 + 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.714 + 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5292259938 + 1.296022750i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5292259938 + 1.296022750i\) |
\(L(1)\) |
\(\approx\) |
\(0.9479843107 + 1.013534740i\) |
\(L(1)\) |
\(\approx\) |
\(0.9479843107 + 1.013534740i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 107 | \( 1 \) |
good | 2 | \( 1 + (0.674 + 0.737i)T \) |
| 3 | \( 1 + (0.0296 + 0.999i)T \) |
| 5 | \( 1 + (0.263 + 0.964i)T \) |
| 7 | \( 1 + (-0.430 - 0.902i)T \) |
| 11 | \( 1 + (0.829 - 0.558i)T \) |
| 13 | \( 1 + (0.582 - 0.812i)T \) |
| 17 | \( 1 + (0.482 - 0.875i)T \) |
| 19 | \( 1 + (-0.320 + 0.947i)T \) |
| 23 | \( 1 + (0.375 + 0.926i)T \) |
| 29 | \( 1 + (0.147 + 0.989i)T \) |
| 31 | \( 1 + (-0.915 - 0.403i)T \) |
| 37 | \( 1 + (0.992 + 0.118i)T \) |
| 41 | \( 1 + (-0.205 + 0.978i)T \) |
| 43 | \( 1 + (0.263 - 0.964i)T \) |
| 47 | \( 1 + (-0.205 - 0.978i)T \) |
| 53 | \( 1 + (0.674 - 0.737i)T \) |
| 59 | \( 1 + (0.147 - 0.989i)T \) |
| 61 | \( 1 + (-0.430 + 0.902i)T \) |
| 67 | \( 1 + (-0.794 - 0.606i)T \) |
| 71 | \( 1 + (0.0296 - 0.999i)T \) |
| 73 | \( 1 + (0.972 - 0.234i)T \) |
| 79 | \( 1 + (0.937 - 0.348i)T \) |
| 83 | \( 1 + (-0.630 - 0.776i)T \) |
| 89 | \( 1 + (-0.717 - 0.696i)T \) |
| 97 | \( 1 + (-0.533 + 0.845i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−29.162605986735645127847482468914, −28.459122286674523060025348184604, −27.91409918641063381747586987042, −25.75450677400696503535886748768, −24.84958786734971775509619549835, −24.05037154905270037142975567383, −23.10631539605663585464164646871, −21.94089397749299465919061056915, −20.93416503973430557292346035059, −19.76339016344628514749835613448, −19.102765954486176309501715859488, −17.95382610802355063182527450912, −16.65575828563143374483010802081, −15.10570873579943192884147790190, −13.95819979492213388274862135904, −12.82112067034759824190970641028, −12.355385483656549918284161788, −11.29572463583725440255090771400, −9.43247318205093175840973625384, −8.65731795296166500450205438487, −6.58573813601668443642991663103, −5.72404664016205691798423736556, −4.23947488172332845425840642442, −2.45377046004863501450960761074, −1.33943407835253767921008958075,
3.23409950137990950903661357763, 3.75988824555213078590662321126, 5.44885584133015591772510092973, 6.483256701307405428368709891326, 7.76723706934372923710659778804, 9.29884663764465266199530893721, 10.53139943547400756053466458874, 11.58099496343850519725192023478, 13.371758569461209737297196278455, 14.24814082062726754330035283036, 15.03378842215647261412888628684, 16.26503435417022408655909980509, 16.93417650213177324134627779475, 18.20760305123965228798161100171, 19.886843731859924000022633970274, 20.99437716698912088766852827577, 22.010709145534190433433606126770, 22.75963665837833998558811909710, 23.43598637574613998894582148443, 25.30982002444757573742697930570, 25.67610721705136046553086857862, 27.01789398694630244084878662981, 27.27436021585686553243454181382, 29.412366066674303243825821930673, 30.005984228238276032927062322683