L(s) = 1 | + (−0.809 − 0.587i)4-s + (−0.309 + 0.951i)9-s + (0.309 + 0.951i)16-s + (−0.618 + 1.90i)31-s + (0.809 − 0.587i)36-s + (0.309 + 0.951i)49-s + (1.61 + 1.17i)59-s + (0.309 − 0.951i)64-s + (0.618 + 1.90i)71-s + (−0.809 − 0.587i)81-s + 2·89-s + ⋯ |
L(s) = 1 | + (−0.809 − 0.587i)4-s + (−0.309 + 0.951i)9-s + (0.309 + 0.951i)16-s + (−0.618 + 1.90i)31-s + (0.809 − 0.587i)36-s + (0.309 + 0.951i)49-s + (1.61 + 1.17i)59-s + (0.309 − 0.951i)64-s + (0.618 + 1.90i)71-s + (−0.809 − 0.587i)81-s + 2·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.550 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.550 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8181253189\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8181253189\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 3 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 7 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 19 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (0.618 - 1.90i)T + (-0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (-1.61 - 1.17i)T + (0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + (-0.618 - 1.90i)T + (-0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 79 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 89 | \( 1 - 2T + T^{2} \) |
| 97 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
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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
−8.885146444164991032101336988537, −8.481499153051907630155629885729, −7.59763942345975854902479078798, −6.75897838605572915177269212883, −5.75363140822042313008061112701, −5.20667232263036746911658833724, −4.50389218712909490844417983709, −3.55680358935366935384333542412, −2.40046962897026905977320897310, −1.25715786843295001244008191271,
0.57309192233742260054144007607, 2.25816433759928509232252633338, 3.42015113556293929284348241979, 3.90714982114245832496713604655, 4.86516480231054871234342287693, 5.70452094168977333896231709206, 6.51733483061489571459384956486, 7.41945413004145304668837427548, 8.110453514753492470841427995430, 8.822774118502955587786298482137