L(s) = 1 | + (−0.707 + 0.707i)7-s + (0.707 + 0.707i)9-s + (1.70 − 0.707i)11-s + (−1 − i)23-s + (−0.707 + 0.707i)25-s + (1.70 + 0.707i)29-s + (0.292 + 0.707i)37-s + (−0.707 + 0.292i)43-s − 1.00i·49-s + (0.707 − 0.292i)53-s − 1.00·63-s + (0.707 + 0.292i)67-s + (−0.707 + 1.70i)77-s + 1.41i·79-s + 1.00i·81-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)7-s + (0.707 + 0.707i)9-s + (1.70 − 0.707i)11-s + (−1 − i)23-s + (−0.707 + 0.707i)25-s + (1.70 + 0.707i)29-s + (0.292 + 0.707i)37-s + (−0.707 + 0.292i)43-s − 1.00i·49-s + (0.707 − 0.292i)53-s − 1.00·63-s + (0.707 + 0.292i)67-s + (−0.707 + 1.70i)77-s + 1.41i·79-s + 1.00i·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 - 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 - 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.358926394\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.358926394\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (0.707 - 0.707i)T \) |
good | 3 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 5 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 11 | \( 1 + (-1.70 + 0.707i)T + (0.707 - 0.707i)T^{2} \) |
| 13 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 23 | \( 1 + (1 + i)T + iT^{2} \) |
| 29 | \( 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2} \) |
| 41 | \( 1 - iT^{2} \) |
| 43 | \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (-0.707 + 0.292i)T + (0.707 - 0.707i)T^{2} \) |
| 59 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 61 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 67 | \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \) |
| 71 | \( 1 - iT^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 - 1.41iT - T^{2} \) |
| 83 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 89 | \( 1 + iT^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.607065375627503786540291949647, −8.373174555377954506761407160403, −7.16441750588894689011853466646, −6.50975507815451510557413332421, −6.01079508288955317946000409826, −4.99501887569020538239016031264, −4.12800688964829162734452772302, −3.37137158014685995945886600951, −2.33279421103037772578154333066, −1.25530732609573155675326774512,
0.953420739465735330293491649141, 2.00534637124756331419519838164, 3.41951711683495666933810492672, 4.03498371271176747974011485737, 4.52660610186374680146439142074, 5.98178813957395323848741532422, 6.48618743494455416457366728523, 7.06669211144725748826969305326, 7.79164654355167949247171506669, 8.796828482429830608201702639802